Formal geometry proofs

1. Proofs of triangle and polygon theorems

Interior and exterior angles of a triangle

In the triangle below, the interior angles are given as M=x, N=60°, and P=60°. As indicated, all three sides of the triangle are the same length. An exterior angle y is formed by extending one side of the triangle toward point Q.

  1. Calculate the size of the interior angle, x.
  2. Calculate the size of the exterior angle, y.
INSTRUCTION: Your answers should be exact (do not round off).
Answer:
  1. x= °
  2. y= °
numeric
numeric
HINT: <no title>
[−1 point ⇒ 1 / 2 points left]

Start with the fact that the interior angles of a triangle sum to 180°.


STEP: Use the sum of angles in a triangle to find x
[−1 point ⇒ 0 / 2 points left]

We need to find two different unknown angles in the diagram. We can find them using facts we know about triangles.

One of the unknown angles, x, is inside of the triangle. So we can use the fact that the interior angles of a triangle have a sum of 180°. Write an equation for this and then solve for x.

Interior angles of a triangle=180°x+60°+60°=180°
x=180°60°60°x=60°
NOTE: You can also find x using the fact that this triangle is equilateral. Equilateral triangles are also equiangular: all three angles are 60°. So even without the equation above we can know that x must be 60°.

Angle x has a measure of 60°.


STEP: Use the exterior angle theorem to find y
[−1 point ⇒ 0 / 2 points left]

The other angle we need to find, y, is outside of the triangle. It is an exterior angle because it is made by the extension of one of the sides of the triangle. To solve for y use the theorem for the exterior angles of a triangle.

The exterior angle theorem for triangles tells us that an exterior angle is equal to the sum of the two interior angles opposite the exterior angle. In this triangle, the exterior angle QM^P is equal to the sum of the opposite interior angles, N and P. The figure below shows this with shaded angles. If we add the orange and blue angles inside the triangle together, we will get an angle which is exactly the same size as the exterior angle.

We can write an equation based on the exterior angle theorem to find the value of y:

Exterior angle of a triangle = The sum of the opposite interior angles

y=N+Py=60°+60°y=120°
NOTE: You can also find y using supplementary angles. The two angles at M must have a sum of 180° because they make a straight line. This approach will lead to the same answer, y=120°.

The correct answers are:

  1. x= 60°
  2. y= 120°

Submit your answer as: and

Interior angles of polygons

Look at the following regular polygon and answer the questions that follow:

Answer: NOTE: Round your answer to two decimal places where necessary.

1. What is the sum of the interior angles of this polygon? °
2. What is the size of each interior angle of this polygon? °

numeric
numeric
STEP: <no title>
[−4 points ⇒ 0 / 4 points left]

The diagram below represents a polygon with 5 sides. This polygon is called a pentagon.

The sum of the interior angles of this pentagon = 540 °.

Let n be the number of sidesn=5Sum of Int angles of a polygon=180(n2)=180×((5)2)=180×(3)Sum of Int angles of a polygon=540°
Each individual angle=540÷5=108°

Submit your answer as: and

Exterior angles of polygons

You are given the following regular polygon with an extension line:

What is the size of the exterior angle x for this polygon?

Answer:

x= °.

numeric
STEP: Calculate the size of the exterior angle
[−2 points ⇒ 0 / 2 points left]

The polygon shown in the question has 5 sides. This is called a pentagon.

We are told that this is a regular polygon. That means that all of the sides have the same length, and all of the angles have the same size.

So we can use the following formula to calculate the size of the exterior angle:

Ext. angle=360°n

where n is the number of sides. So

x=360°5=72°

Submit your answer as:

Problems with polygons

You are given the following regular polygon with a extension line:

What is the size of exterior angle x for the given polygon?

Answer:

x= °

one-of
type(numeric.abserror(0.005))
STEP: <no title>
[−2 points ⇒ 0 / 2 points left]

A plane 2D shape with straight lines that enclose, (one side follows the other side until it forms a closed shape) is a polygon. When all the sides and angles are the same size, it's called a regular polygon. The diagram below represents a polygon with 9 sides. This polygon is called a nonagon.

Let n be the number of sides. Then n=9.

x=Size of Ext angle of polygon=360°n=360°9=40°

We could also calculate the interior angle and then use supplementary rule to find x.

The sum of the interior angles of a polygon is given by 180°(n2).

Therefore the sum of the angles in this polygon is:

180°((9)2)=180°(7)=1 260°

We can use this to work out what each individual interior angle is.

interior angle=1 260°÷9=140°

Finally, we know that the sum of angles on a straight line is 180°. So

exterior angle=180°interior angle=180°140°=40°

So x=40°


Submit your answer as:

Exercises

Interior and exterior angles of a triangle

In the triangle below, the interior angles are given as M=50,98°, N=x, and P=41,75°. An exterior angle y is formed by extending one side of the triangle toward point Q.

  1. Calculate the size of the interior angle, x.
  2. Calculate the size of the exterior angle, y.
INSTRUCTION: Your answers should be exact (do not round off).
Answer:
  1. x= °
  2. y= °
numeric
numeric
HINT: <no title>
[−1 point ⇒ 1 / 2 points left]

Start with the fact that the interior angles of a triangle sum to 180°.


STEP: Use the sum of angles in a triangle to find x
[−1 point ⇒ 0 / 2 points left]

We need to find two different unknown angles in the diagram. We can find them using facts we know about triangles.

One of the unknown angles, x, is inside of the triangle. So we can use the fact that the interior angles of a triangle have a sum of 180°. Write an equation for this and then solve for x.

Interior angles of a triangle=180°50,98°+x+41,75°=180°
x=180°50,98°41,75°x=87,27°

Angle x has a measure of 87,27°.


STEP: Use the exterior angle theorem to find y
[−1 point ⇒ 0 / 2 points left]

The other angle we need to find, y, is outside of the triangle. It is an exterior angle because it is made by the extension of one of the sides of the triangle. To solve for y use the theorem for the exterior angles of a triangle.

The exterior angle theorem for triangles tells us that an exterior angle is equal to the sum of the two interior angles opposite the exterior angle. In this triangle, the exterior angle QM^P is equal to the sum of the opposite interior angles, N and P. The figure below shows this with shaded angles. If we add the orange and blue angles inside the triangle together, we will get an angle which is exactly the same size as the exterior angle.

We can write an equation based on the exterior angle theorem to find the value of y:

Exterior angle of a triangle = The sum of the opposite interior angles

y=N+Py=87,27°+41,75°y=129,02°
NOTE: You can also find y using supplementary angles. The two angles at M must have a sum of 180° because they make a straight line. This approach will lead to the same answer, y=129,02°.

The correct answers are:

  1. x= 87,27°
  2. y= 129,02°

Submit your answer as: and

Interior and exterior angles of a triangle

In the triangle below, the interior angles are given as A=102,12°, B=x, and C=38,94°. Sides AB¯ and AC¯ are equal in length, as indicated. An exterior angle y is formed by extending one side of the triangle toward point D.

  1. Calculate the size of the interior angle, x.
  2. Calculate the size of the exterior angle, y.
INSTRUCTION: Your answers should be exact (do not round off).
Answer:
  1. x= °
  2. y= °
numeric
numeric
HINT: <no title>
[−1 point ⇒ 1 / 2 points left]

Start with the fact that the interior angles of a triangle sum to 180°.


STEP: Use the sum of angles in a triangle to find x
[−1 point ⇒ 0 / 2 points left]

We need to find two different unknown angles in the diagram. We can find them using facts we know about triangles.

One of the unknown angles, x, is inside of the triangle. So we can use the fact that the interior angles of a triangle have a sum of 180°. Write an equation for this and then solve for x.

Interior angles of a triangle=180°102,12°+x+38,94°=180°
x=180°102,12°38,94°x=38,94°
NOTE: You can also solve for x using the fact that this triangle is isosceles (it has two equal sides). Isosceles triangles have two congruent (equal) angles. In this case the equal angles are B and C. And that means that x is equal to 38,94°.

Angle x has a measure of 38,94°.


STEP: Use the exterior angle theorem to find y
[−1 point ⇒ 0 / 2 points left]

The other angle we need to find, y, is outside of the triangle. It is an exterior angle because it is made by the extension of one of the sides of the triangle. To solve for y use the theorem for the exterior angles of a triangle.

The exterior angle theorem for triangles tells us that an exterior angle is equal to the sum of the two interior angles opposite the exterior angle. In this triangle, the exterior angle DA^C is equal to the sum of the opposite interior angles, B and C. The figure below shows this with shaded angles. If we add the orange and blue angles inside the triangle together, we will get an angle which is exactly the same size as the exterior angle.

We can write an equation based on the exterior angle theorem to find the value of y:

Exterior angle of a triangle = The sum of the opposite interior angles

y=B+Cy=38,94°+38,94°y=77,88°
NOTE: You can also find y using supplementary angles. The two angles at A must have a sum of 180° because they make a straight line. This approach will lead to the same answer, y=77,88°.

The correct answers are:

  1. x= 38,94°
  2. y= 77,88°

Submit your answer as: and

Interior and exterior angles of a triangle

In the triangle below, the interior angles are given as M=x, N=43,53°, and P=36,18°. An exterior angle y is formed by extending one side of the triangle toward point Q.

  1. Calculate the size of the interior angle, x.
  2. Calculate the size of the exterior angle, y.
INSTRUCTION: Your answers should be exact (do not round off).
Answer:
  1. x= °
  2. y= °
numeric
numeric
HINT: <no title>
[−1 point ⇒ 1 / 2 points left]

Start with the fact that the interior angles of a triangle sum to 180°.


STEP: Use the sum of angles in a triangle to find x
[−1 point ⇒ 0 / 2 points left]

We need to find two different unknown angles in the diagram. We can find them using facts we know about triangles.

One of the unknown angles, x, is inside of the triangle. So we can use the fact that the interior angles of a triangle have a sum of 180°. Write an equation for this and then solve for x.

Interior angles of a triangle=180°x+43,53°+36,18°=180°
x=180°43,53°36,18°x=100,29°

Angle x has a measure of 100,29°.


STEP: Use the exterior angle theorem to find y
[−1 point ⇒ 0 / 2 points left]

The other angle we need to find, y, is outside of the triangle. It is an exterior angle because it is made by the extension of one of the sides of the triangle. To solve for y use the theorem for the exterior angles of a triangle.

The exterior angle theorem for triangles tells us that an exterior angle is equal to the sum of the two interior angles opposite the exterior angle. In this triangle, the exterior angle QM^P is equal to the sum of the opposite interior angles, N and P. The figure below shows this with shaded angles. If we add the orange and blue angles inside the triangle together, we will get an angle which is exactly the same size as the exterior angle.

We can write an equation based on the exterior angle theorem to find the value of y:

Exterior angle of a triangle = The sum of the opposite interior angles

y=N+Py=43,53°+36,18°y=79,71°
NOTE: You can also find y using supplementary angles. The two angles at M must have a sum of 180° because they make a straight line. This approach will lead to the same answer, y=79,71°.

The correct answers are:

  1. x= 100,29°
  2. y= 79,71°

Submit your answer as: and

Interior angles of polygons

Look at the following regular polygon and answer the questions that follow:

Answer: NOTE: Round your answer to two decimal places where necessary.

1. What is the sum of the interior angles of this polygon? °
2. What is the size of each interior angle of this polygon? °

numeric
numeric
STEP: <no title>
[−4 points ⇒ 0 / 4 points left]

The diagram below represents a polygon with 6 sides. This polygon is called a hexagon.

The sum of the interior angles of this hexagon = 720 °.

Let n be the number of sidesn=6Sum of Int angles of a polygon=180(n2)=180×((6)2)=180×(4)Sum of Int angles of a polygon=720°
Each individual angle=720÷6=120°

Submit your answer as: and

Interior angles of polygons

Look at the following regular polygon and answer the questions that follow:

Answer: NOTE: Round your answer to two decimal places where necessary.

1. What is the sum of the interior angles of this polygon? °
2. What is the size of each interior angle of this polygon? °

numeric
numeric
STEP: <no title>
[−4 points ⇒ 0 / 4 points left]

The diagram below represents a polygon with 3 sides. This polygon is called a triangle.

The sum of the interior angles of this triangle = 180 °.

Let n be the number of sidesn=3Sum of Int angles of a polygon=180(n2)=180×((3)2)=180×(1)Sum of Int angles of a polygon=180°
Each individual angle=180÷3=60°

Submit your answer as: and

Interior angles of polygons

Look at the following regular polygon and answer the questions that follow:

Answer: NOTE: Round your answer to two decimal places where necessary.

1. What is the sum of the interior angles of this polygon? °
2. What is the size of each interior angle of this polygon? °

numeric
numeric
STEP: <no title>
[−4 points ⇒ 0 / 4 points left]

The diagram below represents a polygon with 7 sides. This polygon is called a heptagon.

The sum of the interior angles of this heptagon = 900 °.

Let n be the number of sidesn=7Sum of Int angles of a polygon=180(n2)=180×((7)2)=180×(5)Sum of Int angles of a polygon=900°
Each individual angle=900÷7=128,57°

Submit your answer as: and

Exterior angles of polygons

You are given the following regular polygon with an extension line:

What is the size of the exterior angle x for this polygon?

Answer:

x= °.

numeric
STEP: Calculate the size of the exterior angle
[−2 points ⇒ 0 / 2 points left]

The polygon shown in the question has 8 sides. This is called a ocatagon.

We are told that this is a regular polygon. That means that all of the sides have the same length, and all of the angles have the same size.

So we can use the following formula to calculate the size of the exterior angle:

Ext. angle=360°n

where n is the number of sides. So

x=360°8=45°

Submit your answer as:

Exterior angles of polygons

You are given the following regular polygon with an extension line:

What is the size of the exterior angle x for this polygon?

Answer:

x= °.

numeric
STEP: Calculate the size of the exterior angle
[−2 points ⇒ 0 / 2 points left]

The polygon shown in the question has 3 sides. This is called a triangle.

We are told that this is a regular polygon. That means that all of the sides have the same length, and all of the angles have the same size.

So we can use the following formula to calculate the size of the exterior angle:

Ext. angle=360°n

where n is the number of sides. So

x=360°3=120°

Submit your answer as:

Exterior angles of polygons

You are given the following regular polygon with an extension line:

What is the size of the exterior angle x for this polygon?

Answer:

x= °.

numeric
STEP: Calculate the size of the exterior angle
[−2 points ⇒ 0 / 2 points left]

The polygon shown in the question has 4 sides. This is called a quadrilateral.

We are told that this is a regular polygon. That means that all of the sides have the same length, and all of the angles have the same size.

So we can use the following formula to calculate the size of the exterior angle:

Ext. angle=360°n

where n is the number of sides. So

x=360°4=90°

Submit your answer as:

Problems with polygons

You are given the following regular polygon with a extension line:

What is the size of exterior angle x for the given polygon?

Answer:

x= °

one-of
type(numeric.abserror(0.005))
STEP: <no title>
[−2 points ⇒ 0 / 2 points left]

A plane 2D shape with straight lines that enclose, (one side follows the other side until it forms a closed shape) is a polygon. When all the sides and angles are the same size, it's called a regular polygon. The diagram below represents a polygon with 4 sides. This polygon is called a quadrilateral.

Let n be the number of sides. Then n=4.

x=Size of Ext angle of polygon=360°n=360°4=90°

We could also calculate the interior angle and then use supplementary rule to find x.

The sum of the interior angles of a polygon is given by 180°(n2).

Therefore the sum of the angles in this polygon is:

180°((4)2)=180°(2)=360°

We can use this to work out what each individual interior angle is.

interior angle=360°÷4=90°

Finally, we know that the sum of angles on a straight line is 180°. So

exterior angle=180°interior angle=180°90°=90°

So x=90°


Submit your answer as:

Problems with polygons

You are given the following regular polygon with a extension line:

What is the size of exterior angle x for the given polygon?

Answer:

x= °

one-of
type(numeric.abserror(0.005))
STEP: <no title>
[−2 points ⇒ 0 / 2 points left]

A plane 2D shape with straight lines that enclose, (one side follows the other side until it forms a closed shape) is a polygon. When all the sides and angles are the same size, it's called a regular polygon. The diagram below represents a polygon with 4 sides. This polygon is called a quadrilateral.

Let n be the number of sides. Then n=4.

x=Size of Ext angle of polygon=360°n=360°4=90°

We could also calculate the interior angle and then use supplementary rule to find x.

The sum of the interior angles of a polygon is given by 180°(n2).

Therefore the sum of the angles in this polygon is:

180°((4)2)=180°(2)=360°

We can use this to work out what each individual interior angle is.

interior angle=360°÷4=90°

Finally, we know that the sum of angles on a straight line is 180°. So

exterior angle=180°interior angle=180°90°=90°

So x=90°


Submit your answer as:

Problems with polygons

You are given the following regular polygon with a extension line:

What is the size of exterior angle x for the given polygon?

Answer:

x= °

one-of
type(numeric.abserror(0.005))
STEP: <no title>
[−2 points ⇒ 0 / 2 points left]

A plane 2D shape with straight lines that enclose, (one side follows the other side until it forms a closed shape) is a polygon. When all the sides and angles are the same size, it's called a regular polygon. The diagram below represents a polygon with 4 sides. This polygon is called a quadrilateral.

Let n be the number of sides. Then n=4.

x=Size of Ext angle of polygon=360°n=360°4=90°

We could also calculate the interior angle and then use supplementary rule to find x.

The sum of the interior angles of a polygon is given by 180°(n2).

Therefore the sum of the angles in this polygon is:

180°((4)2)=180°(2)=360°

We can use this to work out what each individual interior angle is.

interior angle=360°÷4=90°

Finally, we know that the sum of angles on a straight line is 180°. So

exterior angle=180°interior angle=180°90°=90°

So x=90°


Submit your answer as:

2. Congruency in triangles

Prove congruency with a common side

In the diagram below, PQSR and PQ=RS=6 cm.

Prove that ΔPQSΔRSQ.

INSTRUCTION: There are often many different ways to prove congruency. You must answer this question by completing the proof below.
Answer:

In ΔPQS and ΔRSQ:

  1. PQ=RS=6 cm (given)

ΔPQSΔRSQ

HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

The two triangles share a side. Which of the four congruency cases could you prove using this common side and the other given information?


STEP: Identify the case for congruency
[−0 points ⇒ 4 / 4 points left]

The four cases for congruency are SSS, SAA, SAS, and 90°HS. When we prove congruency, we must first decide which of the four cases we can prove before writing out the proof.

We were told that PQ=RS=6 cm, so we have a pair of equal sides.

We can see that the line QS is a side of both triangles. When a matching side is shared by two triangles, we say that the side is common. So, we have another pair of equal sides.

We were also told that PQSR, so we can use alternate angles between parallel lines to prove that PQ^S=RS^Q. So we have a pair of equal angles.

The angle is between the two sides, so this matches the SAS (side, angle, side) case for congruency.


STEP: Prove congruency
[−4 points ⇒ 0 / 4 points left]

Now that we have identified the congruency case that can be proven, we need to write out the formal proof. Congruency proofs must always have the same format:

  • State the triangles in which you will prove congruency.
  • Prove the congruency by showing three pairs of matching sides or angles.
  • Conclude, and give a reason for the congruency. Make sure that you label the triangles in the matching order.

In ΔPQS and ΔRSQ:

  1. PQ=RS=6 cm (given)
  2. QS is common
  3. PQ^S=RS^Q (alt s; PQSR)

ΔPQSΔRSQ (SAS).


Submit your answer as: andandand

Congruent triangles

In the diagram below, ΔUVWΔUXW. Also, UWXV while UV=18 and UW=24.

  1. Calculate the value of x.
  2. Determine the length of XV.
Answer:
  1. x= units
  2. XV= units
numeric
numeric
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

The triangles are congruent. That means they are identical. Start by using this fact to label all the sides which are not already labelled.


STEP: Label everything you can in the figure
[−1 point ⇒ 3 / 4 points left]

The first question asks us to determine the length of the side of the diagram labelled x. The x is attached to side WX.

Based on the labels in the diagram, we do not know enough to calculate x immediately. However, the question states that the two triangles are congruent. That means they are identical. We can use that information to label all the sides which are not already labelled. The question also states that UWXV, which means that there are two right angles. We should label those right angles too.

Now we can see two right-angled triangles. And we can see which sides are equal to each other. Both of the triangles include a side labelled x.


STEP: Use the theorem of Pythagoras to find the value of x
[−2 points ⇒ 1 / 4 points left]

Now calculate x using the theorem of Pythagoras. We will focus on triangle UXW, but you can use either one, because they are congruent.

In ΔUXW:UX=18(ΔUVWΔUXW)x2=(18)2+(24)2(Pythagoras)x2=324+576x=±900x=30

The length of WX is 30 units.


STEP: Find the length of XV
[−1 point ⇒ 0 / 4 points left]

For the second question, we need to find the length of segment XV. This segment is made of XU and UV. So we can add them together to get the total length: XV=18+18=36 units.

The correct answers are:

  1. The length of side x is 30 units.
  2. The length of XV is 36 units.

Submit your answer as: and

Deductions from congruency: evaluating proofs

In the diagram below, S^=63°, PR^Q=57°, and SR^P=QP^R=60°.

Prove that RS=PQ.

Langa has already answered the question: his proof is written below. But, he has made a mistake! Look carefully at his proof and identify where he has made his mistake.

Line
one In ΔRSP and ΔPQR:
two 1. SR^P=QP^R=60° (given)
three 2. RPS=PRQ (sum of s in Δ)
four 3. Q^=63° (sum of s in Δ)
five Q^=S^
six ΔRSPΔPQR (SAA)
seven RS=PQ (ΔRSPΔPQR)
Answer:

The mistake is on line .

Replace this line with:

HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

Read through the proof carefully, following every step. Is anything incorrect? Is there anything missing?


STEP: Identify the error in the proof
[−3 points ⇒ 0 / 3 points left]

This option has a mistake in the congruency "proof".

The mistake is on line three: the angles did not have hats on them. But the bigger problem with this proof is that it does not prove a case for congruency.

This option shows that all three angles in the two triangles match up. But, this is a case for similarity: it does not prove congruency. It is possible for two triangles to have the same angles, while being different sizes.

We have to prove congruency in another way. We can use the fact that RP is common to prove SAA.

Line
one In ΔRSP and ΔPQR:
two 1. SR^P=QP^R=60° (given)
three 2. RP is common
four 3. Q^=63° (sum of s in Δ)
five Q^=S^
six ΔRSPΔPQR (SAA)
seven RS=PQ (ΔRSPΔPQR)

Submit your answer as: and

Prove simple congruency

In the diagram below, BE and DA are straight lines that intersect at C. Also, BC^D=90°, BC=EC, and BD=EA.

Prove that ΔBCDΔECA.

INSTRUCTION: There are often many ways to prove congruency. You must answer this question by completing the proof below.
Answer:

In ΔBCD and ΔECA:

  1. BC=EC (given)

HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

The four cases for congruency are SSS, SAA, SAS, and 90°HS. Which of these cases could you prove from the information given?


STEP: Identify the case for congruency
[−0 points ⇒ 4 / 4 points left]

The four cases for congruency are SSS, SAA, SAS, and 90°HS. When we prove congruency, we must first decide which of the four cases we can prove before writing out the proof.

We were given that BC=EC, so we have a pair of equal sides.

We were given that BC^D=90°. We can use vertically opposite angles to prove that BC^D=EC^A=90°. So we have a pair of 90° angles.

We were given that BD=EA, so we have another pair of equal sides. However, these are also the hypotenuses of the triangles, so we have a pair of equal hypotenuses.

This matches the 90°HS (90°, hypotenuse, side) case for congruency.


STEP: Prove congruency
[−4 points ⇒ 0 / 4 points left]

Now that we have identified the congruency case that can be proven, we need to write out the formal proof. Congruency proofs must always have the following format:

  • State the triangles in which you will prove congruency.
  • Prove the congruency by showing three pairs of matching sides or angles. The working to prove each of these three sides or angles is labelled 1, 2, 3.
  • Conclude, and give a reason for the congruency.

In ΔBCD and ΔECA:

  1. BC=EC (given)
  2. BCD=ECA=90° (vert opp s equal)
  3. BD=EA (given)

ΔBCDΔECA (90°HS).

NOTE: You may have thought that we can use SAS because we have two sides and an angle. However, we cannot use this congruency case for this question, because the angle is not between the known sides.

Submit your answer as: andandandandand

Naming congruent triangles

The following two triangles are congruent.

Complete the statement below by naming the second triangle in the correct order, and selecting the correct reason.

INSTRUCTION: You must always label triangles with capital letters. In geometry, "A" and "a" do not mean the same thing.
Answer: ΔABCΔ
string
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Congruent triangles are exactly the same in size and shape. The order in which you write down the vertices of the second triangle must follow the order of the matching vertices in the first triangle. It might not be in alphabetical order!


STEP: Match up the equivalent vertices and sides of the two triangles
[−2 points ⇒ 0 / 2 points left]

You have already been told that the two triangles are congruent. All you need to do is decide what order the vertices of the second triangle should be written in. The equal sides and angles tell you which vertices match up between the two triangles. Write the matching vertices in the same place in the triangle's name.

ΔABCΔ

The first triangle was named for us in the order ABC. This means that the first side named was AB. In our second triangle, the first side we name should be the one that is equal to AB.

Therefore, we will start our second triangle with the letters FD so that side FD will be traced first.

ΔABCΔFD

There is only one vertex (E) left in our second triangle: it will come at the end of the triangle's name.

ΔABCΔFDE

We can see that we have named the second triangle in such a way that the equal sides come in the same order for both triangles.

All three of the pairs of sides were given to be equal, so we use the reason 'side, side, side'. This is abbreviated to 'SSS'.


Submit your answer as: and

Prove congruency with calculations

In the diagram below, K^=73°, ML^J=52°, and KL^M=JM^L=55°

Prove that ΔLKMΔMJL.

INSTRUCTION: Answer this question by selecting the correct proof from the options below. Details are important in proofs, so read carefully.
Answer:
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

To decide on the correct proof, you first need to decide what the proof should look like. In order to prove congruency, you need pairs of equal sides or angles. What geometry fact can you use to determine the third angle in one of the triangles?


STEP: Identify the correct proof
[−4 points ⇒ 0 / 4 points left]

We are looking for a proof that correctly demonstrates one of the four cases for congruency (SSS, SAA, SAS, 90°HS).

Option A

This option correctly proves that all three angles in the two triangles match up. But, this is a case for similarity: it does not prove congruency. It is possible for two triangles to have the same angles, while being different sizes.

Option B

This option correctly demonstrates one of the four cases for congruency. It uses the given information and sum of angles in a triangle to prove one pair of matching sides and two pairs of matching angles (SAA).

Option C

In Step 3, this option says that the reason why K^=J^ is because the two triangles are congruent. Since we haven't proved that the triangles are congruent yet, we cannot use the fact that they are congruent. In Geometry we have to know something for certain before we can use it to prove something else.

Option D

In Step 3, this options says that the reason why LM^K=JL^M=52° is because the two angles are alternate on parallel lines. But, we have not been told that LJKM, so we cannot use this fact (even if it looks like the lines could be parallel). If we want to use it, we have to prove it first. We can only use the information that we have been given, or that we have proven using our geometry facts.


Submit your answer as:

Identifying congruency in triangles

Consider ΔABC below.

The following triangles all look like they might be congruent to ΔABC.

Which of the triangles does not meet the criteria for one of the cases of congruency?

Answer:
HINT: <no title>
[−0 points ⇒ 1 / 1 points left]

There are four cases for congruency: SSS, SAS, SAA, and 90°HS. Which of the options given to you does not satisfy any of these cases, when you compare it to ΔABC?


STEP: Match each triangle with its case for congruency
[−1 point ⇒ 0 / 1 points left]

Although the triangles all look like they might be congruent to ΔABC, we only know for sure that they are congruent if they satisfy one of the cases for congruency.

Triangle 2

Each side of this triangle matches up with one of the sides of ΔABC. Therefore, the two triangles are congruent, because of SSS.

Triangle 3

One of the sides of this triangle matches up with a side of ΔABC. Two of the angles of this triangle match up with angles in ΔABC. Therefore, the two triangles are congruent, because of SAA.

Triangle 4

Two of the sides of this triangle match up to sides of ΔABC, and the angles in between these two sides also match up. Therefore, the two triangles are congruent, because of SAS.

Triangle 1

Although all three of the angles in this triangles are equal to the angles in ΔABC, the two triangles could still have completely different sizes.


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Congruency in circles

In the diagram below, O is the centre of the circle, and K is the midpoint of ML.

Prove that ΔOMKΔOLK.

INSTRUCTION: There are often many ways to prove congruency. You must answer this question by completing the proof below.
Answer:

In ΔOMK and ΔOLK:

  1. MK=LK (given)

ΔOMKΔOLK

HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

The distance from the centre of a circle to its circumference is always the same length. It is called the radius. Which of the four congruency cases could you prove using this fact and the other given information?


STEP: Identify the case for congruency
[−0 points ⇒ 4 / 4 points left]

The four cases for congruency are SSS, SAA, SAS, and 90°HS. When we prove congruency, we must first decide which of the four cases we can prove before writing out the proof.

We were told that MK=LK, so we have a pair of equal sides.

We were also told that O is the centre of the circle. The distance from the centre of a circle to its circumference is always the same length. It is called the radius. We can use this fact to prove that OM=OL, so we have a pair of equal sides.

We can see that the line OK is a side of both triangles. When the same side is shared by two triangles, we say that the side is common. So, we have a pair of equal sides.

This matches the SSS (side, side, side) case for congruency.


STEP: Prove congruency
[−4 points ⇒ 0 / 4 points left]

Now that we have identified the congruency case that can be proven, we need to write out the formal proof. Congruency proofs must always have the following format:

  • State the triangles in which you will prove congruency.
  • Prove the congruency by showing three pairs of matching sides or angles.
  • Conclude, and give a reason for the congruency. Make sure that you label the triangles in the matching order.

In ΔOMK and ΔOLK:

  1. MK=LK (given)
  2. OM=OL (radii)
  3. OK is common

ΔOMKΔOLK (SSS).

NOTE: The reason "radii" is the plural of "radius".

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Consequences of congruency

In this diagram, ΔPQRΔBAC.

Determine the values of x and y, giving reasons for your answers.

NOTE: Diagrams are not necessarily drawn to scale. This means that even if lengths and angles look like they are the same, they might not be equal. You must have a mathematical reason if you say they are equal.
Answer:
  • x= °
  • y= cm
numeric
numeric
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

If two triangles are congruent, they are exactly the same size and shape. What conclusions can you draw about the lengths of sides and sizes of angles in the two triangles?


STEP: Determine which angles and sides are equivalent
[−4 points ⇒ 0 / 4 points left]

If two triangles are congruent, their matching angles and sides will be equal.

We were told that ΔPQRΔBAC, which means that P^ is the same size as B^. In the same way, Q^ is the same size as A^, and R^ is the same size as C^.

This means that x=71°, because P matches B. We use the reason (ΔPQRΔBAC) to remind ourselves that the angles are only equal because the triangles are congruent.

In the same way, PQ is the same length as BA, QR is the same length as AC, and PR is the same length as BC.

This means that y=9 cm, because PQ is equivalent to BA. We use the reason (ΔPQRΔBAC) to remind ourselves that the sides are only equal because the triangles are congruent.


Submit your answer as: andandand

Deductions from congruency

In the diagram below, O is the centre of the circle, and B is the midpoint of EC (in other words, EB=CB). Also, CO^B=52°.

Determine the size of EO^B.

INSTRUCTION: There are often many different ways to answer geometry questions. You must answer this question by completing the proof below.
Answer:

In ΔOEB and ΔOCB:

  1. EB=CB (given)
  2. OE=OC
  3. OB is common

ΔOEBΔOCB

EOB=
EOB= °

numeric
HINT: <no title>
[−0 points ⇒ 5 / 5 points left]

You cannot determine the size of EO^B directly using the geometry facts that you know. The only option is to prove that the two triangles are congruent, and then match up the angles.


STEP: Prove congruency
[−2 points ⇒ 3 / 5 points left]

We cannot determine the size of EO^B directly using the geometry facts that we know. First we must prove that the two triangles are congruent, and then use this fact to match up the angles.

The four cases for congruency are SSS, SAA, SAS, and 90°HS. When we prove congruency, we must first decide which of the four cases we can prove before writing out the proof.

We were told that EB=CB, so we have a pair of equal sides.

We were also told that O is the centre of the circle. The distance from the centre of a circle to its circumference is always the same length. It is called the radius. We can use this fact to prove that OE=OC, so we have a pair of equal sides.

We can see that the line OB is a side of both triangles. When the same side is shared by two triangles, we say that the side is common. So, we have a pair of equal sides.

This matches the SSS (side, side, side) case for congruency.

Now we can write the proof of the congruency:

In ΔOEB and ΔOCB:

  1. EB=CB (given)
  2. OE=OC (radii)
  3. OB is common

ΔOEBΔOCB (SSS)


STEP: Deduce the size of EO^B
[−3 points ⇒ 0 / 5 points left]

The triangles are congruent, so their matching sides and angles are congruent.

EO^B=CO^B(ΔOEBΔOCB)EO^B=52 °
NOTE: We only know that EO^B=CO^B because the triangles are congruent. So, we must write "ΔOEBΔOCB" as the reason for this statement.

Submit your answer as: andandandand

Consequences of congruency with calculations

In this diagram, ΔKJLΔCAB.

Determine the value of y, giving reasons for your answers.

INSTRUCTION: There is more than one step to solve this problem. You should write your final answer for y. Then choose the option which contains all of the reasons that you needed to work it out.
Answer:

y= °

In my working out, I used the following reason(s):

numeric
HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

Since the triangles are congruent, their matching angles will be equal. Can you use a geometry reason to find the size of the angle which matches y?


STEP: Match up the angles of the congruent triangles
[−1 point ⇒ 2 / 3 points left]

If two triangles are congruent, their matching angles and sides will be equal.

We were told that ΔKJLΔCAB which means that K^ is the same size as C^. In the same way, J^ is the same size as A^, and L^ is the same size as B^. The matching sides are also equal, but for this question we only need to think about the angles.

This means that y=K^, because C matches K. We use the reason (ΔKJLΔCAB) to remind ourselves that the angles are only equal because the triangles are congruent.


STEP: Find the value of the missing angle
[−2 points ⇒ 0 / 3 points left]
NOTE: In your working out, the reasons you used must be written next to the line in which you first used the relevant geometry fact. You can't just write all of your reasons at the end. Make sure you read the solution to see where you should have written your reasons.

We don't have enough information to find the size of y in triangle CAB.

But we know that y=K^ and we have enough information to find K^ in triangle KJL, using sum of angles in a triangle.

77°+48°+K^=180°(sum of s in Δ)K^=55°y=55°(ΔKJLΔCAB)

Submit your answer as: and

Identify congruency in overlapping triangles

Consider the diagram below:

In this diagram, PQ=RS=7 units, PS=QR=5 units, PT=TR=3 units, and TS=TQ=6 units.

Identify which triangle is congruent to ΔPSR. Give a reason for the congruency.

INSTRUCTIONS:
  • A full congruency proof is not required for this question.
  • Use upper case letters only.
Answer: ΔPSRΔ
string
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

The triangle you've been given has sides and angles which are equal to sides and angles in another triangle. Use one of the cases for congruency (SSS, SAA, SAS and 90°HS) to match up the triangles.


STEP: Match up equal sides and angles and select the case for congruency
[−2 points ⇒ 0 / 2 points left]

We were given PS=QR and PQ=RS.

PR is a side of ΔPSR and ΔRQP. We say that PR is common.

This means that ΔPSRΔRQP(SSS).

TIP: You must name ΔRQP in that exact order, because that is the order in which the equal angles and sides will match up with ΔPSR.

Submit your answer as: and

Deductions from congruency: overlapping triangles

In the diagram below, QTRS. Also, PQ=PR and QP^T=PR^S.

  1. Complete the congruency proof below:

    Answer:

    In ΔPQT and ΔPRS:

    1. QPT=PRS(given)
    2. PQ=PR(given)

    ΔQTPΔ

    string
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    You are asked to prove congruency in ΔPQT and ΔPRS. How can you use the information given to prove that another pair of sides or angles are equal in these two triangles?


    STEP: Prove another pair of angles equal
    [−2 points ⇒ 2 / 4 points left]

    We have already proved a pair of matching sides and angles. We must prove another pair of sides or angles.

    In ΔPQT and ΔPRS:

    1. QP^T=PR^S(given)
    2. PQ=PR(given)
    3. RS^P=QT^P (corresp s;QTRS)

    STEP: Name the triangles correctly and give a reason for the congruency
    [−2 points ⇒ 0 / 4 points left]

    We have proved the SAA (side, angle, angle) congruency case. We must make sure that we label the triangles in the correct order.

    If we reflect and rotate the triangles, we can see the matching vertices more clearly:

    So, ΔQTPΔPSR (SAA).


    Submit your answer as: andandand
  2. You are now given that RS=25 and QT=30. Hence, determine the length of TS.

    Answer: TS= units
    numeric
    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    TS is the difference between the length of PS and PT. Can you use congruency to match these sides with sides that you know the length of?


    STEP: Use congruency to determine PS and PT
    [−2 points ⇒ 1 / 3 points left]

    PS is made out of PT and TS put together. So PS=PT+TS. If we know PS and PT, we can calculate TS.

    RS=25(given)PT=RS=25(ΔQTPΔPSR)
    QT=30(given)PS=QT=30(ΔQTPΔPSR)

    STEP: Hence, determine TS
    [−1 point ⇒ 0 / 3 points left]
    PS=PT+TS30=25+TSTS=5 units
    NOTE: Although you did not need to give reasons for this question, it is very important to include them in your written out answers. This is so that someone else can understand your thinking. Read our solution carefully to see where each reason should be!

    Submit your answer as:

Deductions from congruency: evaluating proofs

In the diagram below, DABC. Also, BA=13, AD=12, and DC=5.

Prove that BA^D=CA^D.

Akinbode has already answered the question: his proof is written below. But, he has made a mistake! Look carefully at his proof and identify where he has made his mistake.

Line
one In ΔBDA and ΔCDA:
two 1. AC2=122+52 (Pythagoras)
three AC=13
four AC=BA
five 2. BD^A=CD^A=90° (given)
six 3. AD is common
seven ΔBDAΔCDA (90°HS)
eight BA^D=CA^D
Answer:

The mistake is on line .

Replace this line with:

HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

Read through the proof carefully, following every step. Is anything incorrect? Is there anything missing?


STEP: Identify the error in the proof
[−3 points ⇒ 0 / 3 points left]

This option does not have a reason for the deduction on the last line. We know that the angles are equal only because ΔBDAΔCDA. So, we must include this on the last line.

Line
one In ΔBDA and ΔCDA:
two 1. AC2=122+52 (Pythagoras)
three AC=13
four AC=BA
five 2. BD^A=CD^A=90° (given)
six 3. AD is common
seven ΔBDAΔCDA (90°HS)
eight BA^D=CA^D (ΔBDAΔCDA)

Submit your answer as: and

Deductions from congruency: overlapping triangles

In the diagram below, TR=TQ and PQ^R=SR^Q=90°.

  1. Langalibalele needs to prove that ΔPQRΔSRQ. He has already started and his incomplete proof is written below:

    In ΔPQR and ΔSRQ:

    1. PQ^R=SR^Q=90° (given)
    2. QR is common
      ...
    Answer:

    How should Langalibalele complete his proof? Choose the best option.

    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    You are asked to prove that ΔPQRΔSRQ. How can you use the information given to prove that another pair of sides or angles are equal in these two triangles?


    STEP: Complete the proof
    [−3 points ⇒ 0 / 3 points left]

    We need to use the given information to prove that another pair of sides or angles is equal, and hence prove a case for congruency.

    A ✘

    3. PR is the hypotenuse of ΔPQR
    Also, SQ is the hypotenuse of ΔSRQ.
    PR=SQ (hypotenuse with common base)
    ΔPQRΔSRQ (90°HS)

    PR and SQ are hypotenuses of ΔPQR and ΔSRQ respectively. But, this does not mean that they have to be equal in length. Hypotenuse with common base does not refer to any of our geometry theorems. It is just a made up reason. So, this option does not prove that PR=SQ.
    B ✔

    3. TQ^R=TR^Q (s opp equal sides)
    ΔPQRΔSRQ (SAA)

    This option uses geometry reasons correctly to prove that TQ^R=TR^Q. This is a pair of matching angles in both triangles. So this proves that ΔPQRΔSRQ, using SAA.
    C ✘

    3. PT^Q=ST^R (vert opp s)
    ΔPQRΔSRQ (SAA)

    PT^Q is equal to ST^R, because the angles are vertically opposite. But, these are not angles in the triangles that we are trying to prove are congruent.
    D ✘

    3. TQ=TR (given)
    ΔPQRΔSRQ (90°HS)

    TQ is equal to TR. But, these are not sides in the triangles that we are trying to prove are congruent. (They are only parts of the sides in those triangles, which is not enough.)

    The completed proof is shown below. The diagrams are also separated so that the congruency is easier to see.

    In ΔPQR and ΔSRQ:

    1. PQ^R=SR^Q=90° (given)
    2. QR is common
    3. TQ^R=TR^Q (s opp equal sides)

    ΔPQRΔSRQ (SAA)


    Submit your answer as:
  2. You are now given that QR=8 and SQ=10.

    Hence, determine the length of PQ.

    Answer: PQ= units
    numeric
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    Since the triangles are congruent, their matching sides will be equal. Can you use geometry (with reasons) to find the length of the side which matches PQ? Remember that there are right-angled triangles in this diagram.


    STEP: Use the theorem of Pythagoras to calculate SR
    [−3 points ⇒ 1 / 4 points left]

    We do not have enough information to calculate PQ directly. But, in ΔSRQ we do have enough information to calculate the length of SR. Since PQ and SR are matching sides in congruent triangles, they will be equal.

    SQ2=SR2+QR2(Pythagoras)102=SR2+82100=SR2+6410064=SR2+646436=SR2SR2=36SR=36=6 units

    STEP: Hence, determine PQ using congruency
    [−1 point ⇒ 0 / 4 points left]

    We have already proven that the triangles are congruent, so we know that their matching sides and angles are equal.

    PQ=SR(ΔPQRΔSRQ)PQ=6,0 units
    NOTE: Although you did not need to give reasons for this question, it is very important to include them in your written out answers. This is so that someone else can understand your thinking. Read our solution carefully to see where each reason should be!

    Submit your answer as:

Exercises

Prove congruency with a common side

In the diagram below, SR=SQ and SPRQ.

Prove that ΔSRPΔSQP.

INSTRUCTION: There are often many different ways to prove congruency. You must answer this question by completing the proof below.
Answer:

In ΔSRP and ΔSQP:

  1. SR=SQ (given)

ΔSRPΔSQP

HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

The two triangles share a side. Which of the four congruency cases could you prove using this common side and the other given information?


STEP: Identify the case for congruency
[−0 points ⇒ 4 / 4 points left]

The four cases for congruency are SSS, SAA, SAS, and 90°HS. When we prove congruency, we must first decide which of the four cases we can prove before writing out the proof.

We were told that SPRQ, so we know that SP^R=SP^Q=90°. So we have a pair of 90° angles.

We were also given that SR=SQ, so we have a pair of equal sides. However, these are also the hypotenuses of the triangles, so we have a pair of equal hypotenuses.

We can see that the line SP is a side of both triangles. When the same side is shared by the two triangles, we say that the side is common. So, we have a pair of equal sides.

This matches the 90°HS (90°, hypotenuse, side) case for congruency.


STEP: Prove congruency
[−4 points ⇒ 0 / 4 points left]

Now that we have identified the congruency case that can be proven, we need to write out the formal proof. Congruency proofs must always have the same format:

  • State the triangles in which you will prove congruency.
  • Prove the congruency by showing three pairs of matching sides or angles.
  • Conclude, and give a reason for the congruency. Make sure that you label the triangles in the matching order.

In ΔSRP and ΔSQP:

  1. SR=SQ (given)
  2. SP is common
  3. SP^R=SP^Q=90° (given)

ΔSRPΔSQP (90°HS).

NOTE: You might have thought that we can use SAS because we have two sides and an angle. However, we cannot use this congruency case for this question, because the angle is not between the two known sides.

Submit your answer as: andandand

Prove congruency with a common side

In the diagram below, SQRP and SQ=PR=5 mm.

Prove that ΔSQRΔPRQ.

INSTRUCTION: There are often many different ways to prove congruency. You must answer this question by completing the proof below.
Answer:

In ΔSQR and ΔPRQ:

  1. SQ=PR=5 mm (given)

ΔSQRΔPRQ

HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

The two triangles share a side. Which of the four congruency cases could you prove using this common side and the other given information?


STEP: Identify the case for congruency
[−0 points ⇒ 4 / 4 points left]

The four cases for congruency are SSS, SAA, SAS, and 90°HS. When we prove congruency, we must first decide which of the four cases we can prove before writing out the proof.

We were told that SQ=PR=5 mm, so we have a pair of equal sides.

We can see that the line QR is a side of both triangles. When a matching side is shared by two triangles, we say that the side is common. So, we have another pair of equal sides.

We were also told that SQRP, so we can use alternate angles between parallel lines to prove that SQ^R=PR^Q. So we have a pair of equal angles.

The angle is between the two sides, so this matches the SAS (side, angle, side) case for congruency.


STEP: Prove congruency
[−4 points ⇒ 0 / 4 points left]

Now that we have identified the congruency case that can be proven, we need to write out the formal proof. Congruency proofs must always have the same format:

  • State the triangles in which you will prove congruency.
  • Prove the congruency by showing three pairs of matching sides or angles.
  • Conclude, and give a reason for the congruency. Make sure that you label the triangles in the matching order.

In ΔSQR and ΔPRQ:

  1. SQ=PR=5 mm (given)
  2. QR is common
  3. SQ^R=PR^Q (alt s; SQRP)

ΔSQRΔPRQ (SAS).


Submit your answer as: andandand

Prove congruency with a common side

In the diagram below, SRPQ and SR=QP=7 cm.

Prove that ΔSRPΔQPR.

INSTRUCTION: There are often many different ways to prove congruency. You must answer this question by completing the proof below.
Answer:

In ΔSRP and ΔQPR:

  1. SR=QP=7 cm (given)

ΔSRPΔQPR

HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

The two triangles share a side. Which of the four congruency cases could you prove using this common side and the other given information?


STEP: Identify the case for congruency
[−0 points ⇒ 4 / 4 points left]

The four cases for congruency are SSS, SAA, SAS, and 90°HS. When we prove congruency, we must first decide which of the four cases we can prove before writing out the proof.

We were told that SR=QP=7 cm, so we have a pair of equal sides.

We can see that the line RP is a side of both triangles. When a matching side is shared by two triangles, we say that the side is common. So, we have another pair of equal sides.

We were also told that SRPQ, so we can use alternate angles between parallel lines to prove that SR^P=QP^R. So we have a pair of equal angles.

The angle is between the two sides, so this matches the SAS (side, angle, side) case for congruency.


STEP: Prove congruency
[−4 points ⇒ 0 / 4 points left]

Now that we have identified the congruency case that can be proven, we need to write out the formal proof. Congruency proofs must always have the same format:

  • State the triangles in which you will prove congruency.
  • Prove the congruency by showing three pairs of matching sides or angles.
  • Conclude, and give a reason for the congruency. Make sure that you label the triangles in the matching order.

In ΔSRP and ΔQPR:

  1. SR=QP=7 cm (given)
  2. RP is common
  3. SR^P=QP^R (alt s; SRPQ)

ΔSRPΔQPR (SAS).


Submit your answer as: andandand

Congruent triangles

In the diagram below, ΔUVWΔUXW. Also, UWXV while UV=9 and UW=12.

  1. Calculate the value of x.
  2. Determine the length of XV.
Answer:
  1. x= units
  2. XV= units
numeric
numeric
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

The triangles are congruent. That means they are identical. Start by using this fact to label all the sides which are not already labelled.


STEP: Label everything you can in the figure
[−1 point ⇒ 3 / 4 points left]

The first question asks us to determine the length of the side of the diagram labelled x. The x is attached to side WX.

Based on the labels in the diagram, we do not know enough to calculate x immediately. However, the question states that the two triangles are congruent. That means they are identical. We can use that information to label all the sides which are not already labelled. The question also states that UWXV, which means that there are two right angles. We should label those right angles too.

Now we can see two right-angled triangles. And we can see which sides are equal to each other. Both of the triangles include a side labelled x.


STEP: Use the theorem of Pythagoras to find the value of x
[−2 points ⇒ 1 / 4 points left]

Now calculate x using the theorem of Pythagoras. We will focus on triangle UXW, but you can use either one, because they are congruent.

In ΔUXW:UX=9(ΔUVWΔUXW)x2=(9)2+(12)2(Pythagoras)x2=81+144x=±225x=15

The length of WX is 15 units.


STEP: Find the length of XV
[−1 point ⇒ 0 / 4 points left]

For the second question, we need to find the length of segment XV. This segment is made of XU and UV. So we can add them together to get the total length: XV=9+9=18 units.

The correct answers are:

  1. The length of side x is 15 units.
  2. The length of XV is 18 units.

Submit your answer as: and

Congruent triangles

In the diagram below, ΔUVWΔUXW. Also, UWXV while VW=15 and UW=12.

  1. Calculate the value of x.
  2. Determine the length of XV.
Answer:
  1. x= units
  2. XV= units
numeric
numeric
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

The triangles are congruent. That means they are identical. Start by using this fact to label all the sides which are not already labelled.


STEP: Label everything you can in the figure
[−1 point ⇒ 3 / 4 points left]

The first question asks us to determine the length of the side of the diagram labelled x. The x is attached to side UX.

Based on the labels in the diagram, we do not know enough to calculate x immediately. However, the question states that the two triangles are congruent. That means they are identical. We can use that information to label all the sides which are not already labelled. The question also states that UWXV, which means that there are two right angles. We should label those right angles too.

Now we can see two right-angled triangles. And we can see which sides are equal to each other. Both of the triangles include a side labelled x.


STEP: Use the theorem of Pythagoras to find the value of x
[−2 points ⇒ 1 / 4 points left]

Now calculate x using the theorem of Pythagoras. We will focus on triangle UXW, but you can use either one, because they are congruent.

In ΔUXW:XW=15(ΔUVWΔUXW)(15)2=x2+(12)2(Pythagoras)225=x2+144225144=x2±81=x9=x

The length of UX is 9 units.


STEP: Find the length of XV
[−1 point ⇒ 0 / 4 points left]

For the second question, we need to find the length of segment XV. This segment is twice as long as x, which we calculated above. So XV=2(9)=18 units.

The correct answers are:

  1. The length of side x is 9 units.
  2. The length of XV is 18 units.

Submit your answer as: and

Congruent triangles

In the diagram below, ΔUVWΔUXW. Also, UWXV while VW=15 and UW=12.

  1. Calculate the value of x.
  2. Determine the length of XV.
Answer:
  1. x= units
  2. XV= units
numeric
numeric
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

The triangles are congruent. That means they are identical. Start by using this fact to label all the sides which are not already labelled.


STEP: Label everything you can in the figure
[−1 point ⇒ 3 / 4 points left]

The first question asks us to determine the length of the side of the diagram labelled x. The x is attached to side UX.

Based on the labels in the diagram, we do not know enough to calculate x immediately. However, the question states that the two triangles are congruent. That means they are identical. We can use that information to label all the sides which are not already labelled. The question also states that UWXV, which means that there are two right angles. We should label those right angles too.

Now we can see two right-angled triangles. And we can see which sides are equal to each other. Both of the triangles include a side labelled x.


STEP: Use the theorem of Pythagoras to find the value of x
[−2 points ⇒ 1 / 4 points left]

Now calculate x using the theorem of Pythagoras. We will focus on triangle UXW, but you can use either one, because they are congruent.

In ΔUXW:XW=15(ΔUVWΔUXW)(15)2=x2+(12)2(Pythagoras)225=x2+144225144=x2±81=x9=x

The length of UX is 9 units.


STEP: Find the length of XV
[−1 point ⇒ 0 / 4 points left]

For the second question, we need to find the length of segment XV. This segment is twice as long as x, which we calculated above. So XV=2(9)=18 units.

The correct answers are:

  1. The length of side x is 9 units.
  2. The length of XV is 18 units.

Submit your answer as: and

Deductions from congruency: evaluating proofs

In the diagram below, M^=89°, KL^J=41°, and ML^K=JK^L=50°.

Prove that LM=KJ.

Jezile has already answered the question: her proof is written below. Look carefully at her proof and identify where she has made her mistake.

LM=KJ(ΔLMKΔKJL)
Answer:

Jezile's proof is .

Jezile must:

HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

What must you do before assuming that two triangles are congruent?


STEP: Identify the error in the proof
[−3 points ⇒ 0 / 3 points left]

We cannot use congruency to deduce that LM=KJ unless we are certain that the triangles are congruent. Sometimes, we are told in the question that the triangles are congruent. However, this is not the case here. We must first prove that ΔLMKΔKJL before using this fact to answer the question.

Here is the completed proof:

In ΔLMK and ΔKJL:

  1. J^=89° (sum of s in Δ)
    J^=M^
  2. ML^K=JK^L=50° (given)
  3. LK is common

ΔLMKΔKJL (SAA)

LM=KJ (ΔLMKΔKJL)


Submit your answer as: and

Deductions from congruency: evaluating proofs

In the diagram below, J^=66°, KM^L=51°, and JM^K=LK^M=63°.

Prove that MJ=KL.

Adebanjo has already answered the question: his proof is written below. But, he has made a mistake! Look carefully at his proof and identify where he has made his mistake.

Line
one In ΔMJK and ΔKLM:
two 1. JM^K=LK^M=63° (given)
three 2. MK is common
four 3. L^=66° (sum of s in Δ)
five L^=J^
six ΔMJKΔKLM (SAA)
seven MJ=KL
Answer:

The mistake is on line .

Replace this line with:

HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

Read through the proof carefully, following every step. Is anything incorrect? Is there anything missing?


STEP: Identify the error in the proof
[−3 points ⇒ 0 / 3 points left]

This option does not have a reason for the deduction on the last line. We know that the sides are equal only because ΔMJKΔKLM. So, we must include this on the last line.

Line
one In ΔMJK and ΔKLM:
two 1. JM^K=LK^M=63° (given)
three 2. MK is common
four 3. L^=66° (sum of s in Δ)
five L^=J^
six ΔMJKΔKLM (SAA)
seven MJ=KL (ΔMJKΔKLM)

Submit your answer as: and

Deductions from congruency: evaluating proofs

In the diagram below, S^=79°, RP^Q=43°, and SP^R=QR^P=58°.

Prove that PS=RQ.

Halima has already answered the question: her proof is written below. Look carefully at her proof and identify where she has made her mistake.

PS=RQ(ΔPSRΔRQP)
Answer:

Halima's proof is .

Halima must:

HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

What must you do before assuming that two triangles are congruent?


STEP: Identify the error in the proof
[−3 points ⇒ 0 / 3 points left]

We cannot use congruency to deduce that PS=RQ unless we are certain that the triangles are congruent. Sometimes, we are told in the question that the triangles are congruent. However, this is not the case here. We must first prove that ΔPSRΔRQP before using this fact to answer the question.

Here is the completed proof:

In ΔPSR and ΔRQP:

  1. PR is common
  2. Q^=79° (sum of s in Δ)
    Q^=S^
  3. SP^R=QR^P=58° (given)

ΔPSRΔRQP (SAA)

PS=RQ (ΔPSRΔRQP)


Submit your answer as: and

Prove simple congruency

In the diagram below, JK and NM are straight lines that intersect at L. Also, JL^N=90°, JL=KL, and JN=KM.

Prove that ΔJLNΔKLM.

INSTRUCTION: There are often many ways to prove congruency. You must answer this question by completing the proof below.
Answer:

In ΔJLN and ΔKLM:

  1. JL=KL (given)

HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

The four cases for congruency are SSS, SAA, SAS, and 90°HS. Which of these cases could you prove from the information given?


STEP: Identify the case for congruency
[−0 points ⇒ 4 / 4 points left]

The four cases for congruency are SSS, SAA, SAS, and 90°HS. When we prove congruency, we must first decide which of the four cases we can prove before writing out the proof.

We were given that JL=KL, so we have a pair of equal sides.

We were given that JL^N=90°. We can use vertically opposite angles to prove that JL^N=KL^M=90°. So we have a pair of 90° angles.

We were given that JN=KM, so we have another pair of equal sides. However, these are also the hypotenuses of the triangles, so we have a pair of equal hypotenuses.

This matches the 90°HS (90°, hypotenuse, side) case for congruency.


STEP: Prove congruency
[−4 points ⇒ 0 / 4 points left]

Now that we have identified the congruency case that can be proven, we need to write out the formal proof. Congruency proofs must always have the following format:

  • State the triangles in which you will prove congruency.
  • Prove the congruency by showing three pairs of matching sides or angles. The working to prove each of these three sides or angles is labelled 1, 2, 3.
  • Conclude, and give a reason for the congruency.

In ΔJLN and ΔKLM:

  1. JL=KL (given)
  2. JLN=KLM=90° (vert opp s equal)
  3. JN=KM (given)

ΔJLNΔKLM (90°HS).

NOTE: You may have thought that we can use SAS because we have two sides and an angle. However, we cannot use this congruency case for this question, because the angle is not between the known sides.

Submit your answer as: andandandandand

Prove simple congruency

In the diagram below, TP and RS are straight lines that intersect at Q. Also, TQ=SQ and QR=QP.

Prove that ΔTQRΔSQP.

INSTRUCTION: There are often many ways to prove congruency. You must answer this question by completing the proof below.
Answer:

In ΔTQR and ΔSQP:

  1. TQ=SQ (given)

HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

The four cases for congruency are SSS, SAA, SAS, and 90°HS. Which of these cases could you prove from the information given?


STEP: Identify the case for congruency
[−0 points ⇒ 4 / 4 points left]

The four cases for congruency are SSS, SAA, SAS, and 90°HS. When we prove congruency, we must first decide which of the four cases we can prove before writing out the proof.

We were given that TQ=SQ, so we have a pair of equal sides.

We were given that QR=QP, so we have another pair of equal sides.

We can use vertically opposite angles to prove that TQ^R=SQ^P. So we have a pair of equal angles.

The angle is between the two sides, so this matches the SAS (side, angle, side) case for congruency.


STEP: Prove congruency
[−4 points ⇒ 0 / 4 points left]

Now that we have identified the congruency case that can be proven, we need to write out the formal proof. Congruency proofs must always have the following format:

  • State the triangles in which you will prove congruency.
  • Prove the congruency by showing three pairs of matching sides or angles. The working to prove each of these three sides or angles is labelled 1, 2, 3.
  • Conclude, and give a reason for the congruency.

In ΔTQR and ΔSQP:

  1. TQ=SQ (given)
  2. QR=QP (given)
  3. TQR=SQP (vert opp s equal)

ΔTQRΔSQP (SAS).


Submit your answer as: andandandandand

Prove simple congruency

In the diagram below, JM and KL are straight lines that intersect at N. Also, JKLM and JN=MN.

Prove that ΔJNKΔMNL.

INSTRUCTION: There are often many ways to prove congruency. You must answer this question by completing the proof below.
Answer:

In ΔJNK and ΔMNL:

  1. JN=MN (given)

HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

The four cases for congruency are SSS, SAA, SAS, and 90°HS. Which of these cases could you prove from the information given?


STEP: Identify the case for congruency
[−0 points ⇒ 4 / 4 points left]

The four cases for congruency are SSS, SAA, SAS, and 90°HS. When we prove congruency, we must first decide which of the four cases we can prove before writing out the proof.

We were given that JN=MN, so we have a pair of equal sides.

We were also told that JKLM, so we can use alternate angles between parallel lines to prove that J^=M^. So we have a pair of equal angles.

We can use vertically opposite angles to prove that JN^L=MN^L. So we have another pair of equal angles.

This matches the SAA (side, angle, angle) case for congruency.


STEP: Prove congruency
[−4 points ⇒ 0 / 4 points left]

Now that we have identified the congruency case that can be proven, we need to write out the formal proof. Congruency proofs must always have the following format:

  • State the triangles in which you will prove congruency.
  • Prove the congruency by showing three pairs of matching sides or angles. The working to prove each of these three sides or angles is labelled 1, 2, 3.
  • Conclude, and give a reason for the congruency.

In ΔJNK and ΔMNL:

  1. JN=MN (given)
  2. J=M (alt s; JKLM)
  3. JNK=MNL (vert opp s equal)

ΔJNKΔMNL (SAA).

NOTE: You could also have proved that K=M using alternate angles. But, you had to answer this question by choosing from the options, and this combination was not an option.

Submit your answer as: andandandandand

Naming congruent triangles

The following two triangles are congruent.

Complete the statement below by naming the second triangle in the correct order, and selecting the correct reason.

INSTRUCTION: You must always label triangles with capital letters. In geometry, "A" and "a" do not mean the same thing.
Answer: ΔABCΔ
string
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Congruent triangles are exactly the same in size and shape. The order in which you write down the vertices of the second triangle must follow the order of the matching vertices in the first triangle. It might not be in alphabetical order!


STEP: Match up the equivalent vertices and sides of the two triangles
[−2 points ⇒ 0 / 2 points left]

You have already been told that the two triangles are congruent. All you need to do is decide what order the vertices of the second triangle should be written in. The equal sides and angles tell you which vertices match up between the two triangles. Write the matching vertices in the same place in the triangle's name.

ΔABCΔ

The first triangle was named for us in the order ABC. We can see that the angle B^ is equal to angle F^. Therefore B and F must be in the same position in the triangle's names.

ΔABCΔF

We can also see that the side AB is equal in length to the side DF. Therefore, A must be in the same position as D.

ΔABCΔDF

There is only one vertex (E) left in our second triangle: it will come at the end of the triangle's name.

ΔABCΔDFE

We can see that we have named the second triangle in such a way that the equal sides come in the same order for both triangles.

Two of the sides were given to be equal, as well as the angle in between the given sides. We use the reason 'side, angle, side', which is abbreviated to 'SAS'.


Submit your answer as: and

Naming congruent triangles

The following two triangles are congruent.

Complete the statement below by naming the second triangle in the correct order, and selecting the correct reason.

INSTRUCTION: You must always label triangles with capital letters. In geometry, "A" and "a" do not mean the same thing.
Answer: ΔABCΔ
string
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Congruent triangles are exactly the same in size and shape. The order in which you write down the vertices of the second triangle must follow the order of the matching vertices in the first triangle. It might not be in alphabetical order!


STEP: Match up the equivalent vertices and sides of the two triangles
[−2 points ⇒ 0 / 2 points left]

You have already been told that the two triangles are congruent. All you need to do is decide what order the vertices of the second triangle should be written in. The equal sides and angles tell you which vertices match up between the two triangles. Write the matching vertices in the same place in the triangle's name.

ΔABCΔ

The first triangle was named for us in the order ABC. We can see that angle C^ is equal to angle F^, because they are both 90°. Therefore C and F must be in the position in the triangles' names.

ΔABCΔF

We can also see that side BC is equal in length to DF. Therefore, we must place the letter D in the same position as B.

ΔABCΔDF

There is only one vertex (E) left in our second triangle: it will fill in the last empty space.

ΔABCΔEDF

We can see that we have named the triangle in such a way that the equal angles and equal sides come in the same order for both triangles.

A 90° angle was given in both triangles, together with the hypotenuse and one other side. We use the reason '90°, hypotenuse, side', which is abbreviated to '90°HS'.


Submit your answer as: and

Naming congruent triangles

The following two triangles are congruent.

Complete the statement below by naming the second triangle in the correct order, and selecting the correct reason.

INSTRUCTION: You must always label triangles with capital letters. In geometry, "A" and "a" do not mean the same thing.
Answer: ΔABCΔ
string
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Congruent triangles are exactly the same in size and shape. The order in which you write down the vertices of the second triangle must follow the order of the matching vertices in the first triangle. It might not be in alphabetical order!


STEP: Match up the equivalent vertices and sides of the two triangles
[−2 points ⇒ 0 / 2 points left]

You have already been told that the two triangles are congruent. All you need to do is decide what order the vertices of the second triangle should be written in. The equal sides and angles tell you which vertices match up between the two triangles. Write the matching vertices in the same place in the triangle's name.

ΔABCΔ

The first triangle was named for us in the order ABC. We can see that angle B^ is equal to angle Y^. Therefore, B and Y must be in the same position in the triangles' names. In the same way, angle C^ is equal to Z^. Therefore C and Z must be in the same position in the triangles' names.

ΔABCΔYZ

There is only one vertex (X) left in our second triangle: it will fill in the last empty space.

ΔABCΔXYZ

We can see that we have named the second triangle in such a way that the equal angles and equal sides come in the same order for both triangles.

One side and two angles were given to be equal, so we use the reason 'side, angle, angle'. This is abbreviated to 'SAA'.


Submit your answer as: and

Prove congruency with calculations

In the diagram below, J^=69°, MK^L=54°, and JK^M=LM^K=57°

Prove that ΔKJMΔMLK.

INSTRUCTION: Answer this question by selecting the correct proof from the options below. Details are important in proofs, so read carefully.
Answer:
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

To decide on the correct proof, you first need to decide what the proof should look like. In order to prove congruency, you need pairs of equal sides or angles. What geometry fact can you use to determine the third angle in one of the triangles?


STEP: Identify the correct proof
[−4 points ⇒ 0 / 4 points left]

We are looking for a proof that correctly demonstrates one of the four cases for congruency (SSS, SAA, SAS, 90°HS).

Option A

In Step 2, this options says that the reason why KM^J=LK^M=54° is because the two angles are alternate on parallel lines. But, we have not been told that KLJM, so we cannot use this fact (even if it looks like the lines could be parallel). If we want to use it, we have to prove it first. We can only use the information that we have been given, or that we have proven using our geometry facts.

Option B

In Step 2, this option says that the reason why J^=L^ is because the two triangles are congruent. Since we haven't proved that the triangles are congruent yet, we cannot use the fact that they are congruent. In Geometry we have to know something for certain before we can use it to prove something else.

Option C

This option correctly demonstrates one of the four cases for congruency. It uses the given information and sum of angles in a triangle to prove one pair of matching sides and two pairs of matching angles (SAA).

Option D

This option correctly proves that all three angles in the two triangles match up. But, this is a case for similarity: it does not prove congruency. It is possible for two triangles to have the same angles, while being different sizes.


Submit your answer as:

Prove congruency with calculations

In the diagram below, B^=71°, DA^C=41°, and BA^D=CD^A=68°

Prove that ΔABDΔDCA.

INSTRUCTION: Answer this question by selecting the correct proof from the options below. Details are important in proofs, so read carefully.
Answer:
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

To decide on the correct proof, you first need to decide what the proof should look like. In order to prove congruency, you need pairs of equal sides or angles. What geometry fact can you use to determine the third angle in one of the triangles?


STEP: Identify the correct proof
[−4 points ⇒ 0 / 4 points left]

We are looking for a proof that correctly demonstrates one of the four cases for congruency (SSS, SAA, SAS, 90°HS).

Option A

This option correctly proves that all three angles in the two triangles match up. But, this is a case for similarity: it does not prove congruency. It is possible for two triangles to have the same angles, while being different sizes.

Option B

In Step 2, this options says that the reason why AD^B=CA^D=41° is because the two angles are alternate on parallel lines. But, we have not been told that ACBD, so we cannot use this fact (even if it looks like the lines could be parallel). If we want to use it, we have to prove it first. We can only use the information that we have been given, or that we have proven using our geometry facts.

Option C

In Step 2, this option says that the reason why B^=C^ is because the two triangles are congruent. Since we haven't proved that the triangles are congruent yet, we cannot use the fact that they are congruent. In Geometry we have to know something for certain before we can use it to prove something else.

Option D

This option correctly demonstrates one of the four cases for congruency. It uses the given information and sum of angles in a triangle to prove one pair of matching sides and two pairs of matching angles (SAA).


Submit your answer as:

Prove congruency with calculations

In the diagram below, SPRQ. Also, RP=10, PS=8, and SQ=6.

Prove that ΔRSPΔQSP.

INSTRUCTION: Answer this question by selecting the correct proof from the options below. Details are important in proofs, so read carefully.
Answer:
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

To decide on the correct proof, you first need to decide what the proof should look like. In order to prove congruency, you need pairs of equal sides or angles. What geometry fact can you use to determine the third side in one of the right-angled triangles?


STEP: Identify the correct proof
[−4 points ⇒ 0 / 4 points left]

We are looking for a proof that correctly demonstrates one of the four cases for congruency (SSS, SAA, SAS, 90°HS).

Option A

In Step 2, this option says that P^ is common. But, a common angle is an angle that is shared by two triangles. In this case, there are two different angles at Point P: RP^S and QP^S. They might be equal, or they might not. They are definitely not common, as each triangle has its own angle at P.

Option B

This proof does not include a reason for the congruency on the last line, so it cannot be the correct choice.

Option C

In Step 3, this option says that the reason why PQ=10 is because the two triangles are congruent. But, this is what we are trying to prove. We cannot use it as a fact to prove itself.

Option D

This option correctly demonstrates one of the four cases for congruency. It uses the given information and the theorem of Pythagoras to prove one pair of matching 90° angles, one pair of hypotenuses, and one pair of matching sides (90°HS).

NOTE: You could also have proved that the triangles were congruent by calculating RS and using SAS.

Submit your answer as:

Identifying congruency in triangles

Consider ΔDEF below.

The following triangles all look like they might be congruent to ΔDEF.

Which of the triangles does not meet the criteria for one of the cases of congruency?

Answer:
HINT: <no title>
[−0 points ⇒ 1 / 1 points left]

There are four cases for congruency: SSS, SAS, SAA, and 90°HS. Which of the options given to you does not satisfy any of these cases, when you compare it to ΔDEF?


STEP: Match each triangle with its case for congruency
[−1 point ⇒ 0 / 1 points left]

Although the triangles all look like they might be congruent to ΔDEF, we only know for sure that they are congruent if they satisfy one of the cases for congruency.

Triangle 1

Two of the sides of this triangle match up to sides of ΔDEF, and the angles in between these two sides also match up. Therefore, the two triangles are congruent, because of SAS.

Triangle 2

Each side of this triangle matches up with one of the sides of ΔDEF. Therefore, the two triangles are congruent, because of SSS.

Triangle 3

One of the sides of this triangle matches up with a side of ΔDEF. Two of the angles of this triangle match up with angles in ΔDEF. Therefore, the two triangles are congruent, because of SAA.

Triangle 4

Although this triangle has one side and one angle which match up to a side and angle in ΔDEF, this is not enough to know for sure that the triangles are congruent. We need at least one more side or angle pair.


Submit your answer as:

Identifying congruency in triangles

Consider ΔABC below.

The following triangles all look like they might be congruent to ΔABC.

Which of the triangles does not meet the criteria for one of the cases of congruency?

Answer:
HINT: <no title>
[−0 points ⇒ 1 / 1 points left]

There are four cases for congruency: SSS, SAS, SAA, and 90°HS. Which of the options given to you does not satisfy any of these cases, when you compare it to ΔABC?


STEP: Match each triangle with its case for congruency
[−1 point ⇒ 0 / 1 points left]

Although the triangles all look like they might be congruent to ΔABC, we only know for sure that they are congruent if they satisfy one of the cases for congruency.

Triangle 1

Two of the sides of this triangle match up to sides of ΔABC, and the angles in between these two sides also match up. Therefore, the two triangles are congruent, because of SAS.

Triangle 2

One of the sides of this triangle matches up with a side of ΔABC. Two of the angles of this triangle match up with angles in ΔABC. Therefore, the two triangles are congruent, because of SAA.

Triangle 4

Each side of this triangle matches up with one of the sides of ΔABC. Therefore, the two triangles are congruent, because of SSS.

Triangle 3

Although all three of the angles in this triangles are equal to the angles in ΔABC, the two triangles could still have completely different sizes.


Submit your answer as:

Identifying congruency in triangles

Consider ΔABC below.

The following triangles all look like they might be congruent to ΔABC.

Which of the triangles does not meet the criteria for one of the cases of congruency?

Answer:
HINT: <no title>
[−0 points ⇒ 1 / 1 points left]

There are four cases for congruency: SSS, SAS, SAA, and 90°HS. Which of the options given to you does not satisfy any of these cases, when you compare it to ΔABC?


STEP: Match each triangle with its case for congruency
[−1 point ⇒ 0 / 1 points left]

Although the triangles all look like they might be congruent to ΔABC, we only know for sure that they are congruent if they satisfy one of the cases for congruency.

Triangle 2

Each side of this triangle matches up with one of the sides of ΔABC. Therefore, the two triangles are congruent, because of SSS.

Triangle 3

One of the sides of this triangle matches up with a side of ΔABC. Two of the angles of this triangle match up with angles in ΔABC. Therefore, the two triangles are congruent, because of SAA.

Triangle 4

Two of the sides of this triangle match up to sides of ΔABC, and the angles in between these two sides also match up. Therefore, the two triangles are congruent, because of SAS.

Triangle 1

Although all three of the angles in this triangles are equal to the angles in ΔABC, the two triangles could still have completely different sizes.


Submit your answer as:

Congruency in circles

In the diagram below, O is the centre of the circle, and OC=OB. Also, EB and CD are straight lines.

Prove that ΔOECΔODB.

INSTRUCTION: There are often many ways to prove congruency. You must answer this question by completing the proof below.
Answer:

In ΔOEC and ΔODB:

  1. OC=OB (given)
  2. EOC=DOB

ΔOECΔODB

HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

The distance from the centre of a circle to its circumference is always the same length. It is called the radius. Which of the four congruency cases could you prove using this fact and the other given information?


STEP: Identify the case for congruency
[−0 points ⇒ 4 / 4 points left]

The four cases for congruency are SSS, SAA, SAS, and 90°HS. When we prove congruency, we must first decide which of the four cases we can prove before writing out the proof.

We were told that OC=OB, so we have a pair of equal sides.

We were also told that EB and CD are straight lines, so they form vertically opposite angles where they intersect. We can use this fact to prove that EO^C=DO^B . So, we have a pair of equal angles.

Finally, we were told that O is the centre of the circle. The distance from the centre of a circle to its circumference is always the same length. It is called the radius. We can use this fact to prove that OE=OD. So, we have another pair of equal sides.

The angle is between the two sides, so this matches the SAS (side, angle, side) case for congruency.


STEP: Prove congruency
[−4 points ⇒ 0 / 4 points left]

Now that we have identified the congruency case that can be proven, we need to write out the formal proof. Congruency proofs must always have the following format:

  • State the triangles in which you will prove congruency.
  • Prove the congruency by showing three pairs of matching sides or angles.
  • Conclude, and give a reason for the congruency. Make sure that you label the triangles in the matching order.

In ΔOEC and ΔODB:

  1. OC=OB (given)
  2. EO^C=DO^B (vert opp s equal)
  3. OE=OD (radii)

ΔOECΔODB (SAS).

NOTE: The reason "radii" is the plural of "radius".

Submit your answer as: andandand

Congruency in circles

In the diagram below, O is the centre of the circle, and M is the midpoint of KJ.

Prove that ΔOKMΔOJM.

INSTRUCTION: There are often many ways to prove congruency. You must answer this question by completing the proof below.
Answer:

In ΔOKM and ΔOJM:

  1. KM=JM (given)

ΔOKMΔOJM

HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

The distance from the centre of a circle to its circumference is always the same length. It is called the radius. Which of the four congruency cases could you prove using this fact and the other given information?


STEP: Identify the case for congruency
[−0 points ⇒ 4 / 4 points left]

The four cases for congruency are SSS, SAA, SAS, and 90°HS. When we prove congruency, we must first decide which of the four cases we can prove before writing out the proof.

We were told that KM=JM, so we have a pair of equal sides.

We were also told that O is the centre of the circle. The distance from the centre of a circle to its circumference is always the same length. It is called the radius. We can use this fact to prove that OK=OJ, so we have a pair of equal sides.

We can see that the line OM is a side of both triangles. When the same side is shared by two triangles, we say that the side is common. So, we have a pair of equal sides.

This matches the SSS (side, side, side) case for congruency.


STEP: Prove congruency
[−4 points ⇒ 0 / 4 points left]

Now that we have identified the congruency case that can be proven, we need to write out the formal proof. Congruency proofs must always have the following format:

  • State the triangles in which you will prove congruency.
  • Prove the congruency by showing three pairs of matching sides or angles.
  • Conclude, and give a reason for the congruency. Make sure that you label the triangles in the matching order.

In ΔOKM and ΔOJM:

  1. KM=JM (given)
  2. OK=OJ (radii)
  3. OM is common

ΔOKMΔOJM (SSS).

NOTE: The reason "radii" is the plural of "radius".

Submit your answer as: andandand

Congruency in circles

In the diagram below, O is the centre of the circle, and ORPQ.

Prove that ΔOPRΔOQR.

INSTRUCTION: There are often many ways to prove congruency. You must answer this question by completing the proof below.
Answer:

In ΔOPR and ΔOQR:

  1. ORP=ORQ=90° (given)

ΔOPRΔOQR

HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

The distance from the centre of a circle to its circumference is always the same length. It is called the radius. Which of the four congruency cases could you prove using this fact and the other given information?


STEP: Identify the case for congruency
[−0 points ⇒ 4 / 4 points left]

The four cases for congruency are SSS, SAA, SAS, and 90°HS. When we prove congruency, we must first decide which of the four cases we can prove before writing out the proof.

We were told that ORPQ, so we know that OR^P=OR^Q=90°. So we have a pair of 90° angles.

We were also told that O is the centre of the circle. The distance from the centre of a circle to its circumference is always the same length. It is called the radius. We can use this fact to prove that OP=OQ, so we have a pair of equal sides. However, OP is the hypotenuse of ΔOPR, and OQ is the hypotenuse of ΔOQR, because they are opposite the 90° angles. So we have a pair of equal hypotenuses.

We can see that the line OR is a side of both triangles. When the same side is shared by two triangles, we say that the side is common. So, we have a pair of equal sides.

This matches the 90°HS (90°, hypotenuse, side) case for congruency.


STEP: Prove congruency
[−4 points ⇒ 0 / 4 points left]

Now that we have identified the congruency case that can be proven, we need to write out the formal proof. Congruency proofs must always have the following format:

  • State the triangles in which you will prove congruency.
  • Prove the congruency by showing three pairs of matching sides or angles.
  • Conclude, and give a reason for the congruency. Make sure that you label the triangles in the matching order.

In ΔOPR and ΔOQR:

  1. OR^P=OR^Q=90° (given)
  2. OP=OQ (radii)
  3. OR is common

ΔOPRΔOQR (90°HS).

NOTE: The reason "radii" is the plural of "radius".

Submit your answer as: andandand

Consequences of congruency

In this diagram, ΔKLJΔABC.

Determine the values of x and y, giving reasons for your answers.

NOTE: Diagrams are not necessarily drawn to scale. This means that even if lengths and angles look like they are the same, they might not be equal. You must have a mathematical reason if you say they are equal.
Answer:
  • x= °
  • y= units
numeric
numeric
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

If two triangles are congruent, they are exactly the same size and shape. What conclusions can you draw about the lengths of sides and sizes of angles in the two triangles?


STEP: Determine which angles and sides are equivalent
[−4 points ⇒ 0 / 4 points left]

If two triangles are congruent, their matching angles and sides will be equal.

We were told that ΔKLJΔABC, which means that K^ is the same size as A^. In the same way, L^ is the same size as B^, and J^ is the same size as C^.

This means that x=54°, because K matches A. We use the reason (ΔKLJΔABC) to remind ourselves that the angles are only equal because the triangles are congruent.

In the same way, KL is the same length as AB, LJ is the same length as BC, and KJ is the same length as AC.

This means that y=9,76 units, because JK is equivalent to CA. We use the reason (ΔKLJΔABC) to remind ourselves that the sides are only equal because the triangles are congruent.


Submit your answer as: andandand

Consequences of congruency

In this diagram, ΔYZXΔJKL.

Determine the values of a and b, giving reasons for your answers.

NOTE: Diagrams are not necessarily drawn to scale. This means that even if lengths and angles look like they are the same, they might not be equal. You must have a mathematical reason if you say they are equal.
Answer:
  • a= °
  • b= m
numeric
numeric
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

If two triangles are congruent, they are exactly the same size and shape. What conclusions can you draw about the lengths of sides and sizes of angles in the two triangles?


STEP: Determine which angles and sides are equivalent
[−4 points ⇒ 0 / 4 points left]

If two triangles are congruent, their matching angles and sides will be equal.

We were told that ΔYZXΔJKL, which means that Y^ is the same size as J^. In the same way, Z^ is the same size as K^, and X^ is the same size as L^.

This means that a=52°, because X matches L. We use the reason (ΔYZXΔJKL) to remind ourselves that the angles are only equal because the triangles are congruent.

In the same way, YZ is the same length as JK, ZX is the same length as KL, and YX is the same length as JL.

This means that b=17,35 m, because XY is equivalent to LJ. We use the reason (ΔYZXΔJKL) to remind ourselves that the sides are only equal because the triangles are congruent.


Submit your answer as: andandand

Consequences of congruency

In this diagram, ΔPQRΔJKL.

Determine the values of a and b, giving reasons for your answers.

NOTE: Diagrams are not necessarily drawn to scale. This means that even if lengths and angles look like they are the same, they might not be equal. You must have a mathematical reason if you say they are equal.
Answer:
  • a= °
  • b= m
numeric
numeric
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

If two triangles are congruent, they are exactly the same size and shape. What conclusions can you draw about the lengths of sides and sizes of angles in the two triangles?


STEP: Determine which angles and sides are equivalent
[−4 points ⇒ 0 / 4 points left]

If two triangles are congruent, their matching angles and sides will be equal.

We were told that ΔPQRΔJKL, which means that P^ is the same size as J^. In the same way, Q^ is the same size as K^, and R^ is the same size as L^.

This means that a=42°, because R matches L. We use the reason (ΔPQRΔJKL) to remind ourselves that the angles are only equal because the triangles are congruent.

In the same way, PQ is the same length as JK, QR is the same length as KL, and PR is the same length as JL.

This means that b=8,49 m, because PQ is equivalent to JK. We use the reason (ΔPQRΔJKL) to remind ourselves that the sides are only equal because the triangles are congruent.


Submit your answer as: andandand

Deductions from congruency

In the diagram below, O is the centre of the circle, and OC=OD. Also, ED and CB are straight lines, BD=12 units, and C^=79°.

Determine the size of D^.

INSTRUCTION: There are often many different ways to answer geometry questions. You must answer this question by completing the proof below.
Answer:

In ΔOEC and ΔOBD:

  1. OC=OD (given)
  2. EOC=BOD (vert opp s equal)

ΔOECΔOBD (SAS)

D=
D= °

numeric
HINT: <no title>
[−0 points ⇒ 5 / 5 points left]

You cannot determine the size of D^ directly using the geometry facts that you know. The only option is to prove that the two triangles are congruent, and then match up the angles.


STEP: Prove congruency
[−2 points ⇒ 3 / 5 points left]

We cannot determine the size of D^ directly using the geometry facts that we know. First we must prove that the two triangles are congruent, and then use this fact to match up the angles.

The four cases for congruency are SSS, SAA, SAS, and 90°HS. When we prove congruency, we must first decide which of the four cases we can prove before writing out the proof.

We were told that OC=OD, so we have a pair of equal sides.

We were also told that ED and CB are straight lines, so they form vertically opposite angles where they intersect. We can use this fact to prove that EO^C=BO^D . So, we have a pair of equal angles.

Finally, we were told that O is the centre of the circle. The distance from the centre of a circle to its circumference is always the same length. It is called the radius. We can use this fact to prove that OE=OB. So, we have another pair of equal sides.

The angle is between the two sides, so this matches the SAS (side, angle, side) case for congruency.

Now we can write the proof of the congruency:

In ΔOEC and ΔOBD:

  1. OC=OD (given)
  2. EO^C=BO^D (vert opp s equal)
  3. OE=OB (radii)

ΔOECΔOBD (SAS)


STEP: Deduce the size of D^
[−3 points ⇒ 0 / 5 points left]

The triangles are congruent, so their matching sides and angles are congruent.

D^=C^(ΔOBEΔOCD)D^=79 °
NOTE: We only know that D^=C^ because the triangles are congruent. So, we must write "ΔOBEΔOCD" as the reason for this statement.

Submit your answer as: andandandand

Deductions from congruency

In the diagram below, O is the centre of the circle, and OECB. Also, CE=3 units and BO^E=42°.

Determine the length of BE.

INSTRUCTION: There are often many different ways to answer geometry questions. You must answer this question by completing the proof below.
Answer:

In ΔOCE and ΔOBE:

  1. OEC=OEB=90° (given)
  2. OC=OB
  3. OE is common

ΔOCEΔOBE

BE=
BE= units

numeric
HINT: <no title>
[−0 points ⇒ 5 / 5 points left]

You cannot determine the length of BE directly using the geometry facts that you know. The only option is to prove that the two triangles are congruent, and then match up the sides.


STEP: Prove congruency
[−2 points ⇒ 3 / 5 points left]

We cannot determine the length of BE directly using the geometry facts that we know. First we must prove that the two triangles are congruent, and then use this fact to match up the sides.

The four cases for congruency are SSS, SAA, SAS, and 90°HS. When we prove congruency, we must first decide which of the four cases we can prove before writing out the proof.

We were told that OECB, so we know that OE^C=OE^B=90°. So we have a pair of 90° angles.

We were also told that O is the centre of the circle. The distance from the centre of a circle to its circumference is always the same length. It is called the radius. We can use this fact to prove that OC=OB, so we have a pair of equal sides. However, OC is the hypotenuse of ΔOCE, and OB is the hypotenuse of ΔOBE, because they are opposite the 90° angles. So we have a pair of equal hypotenuses.

We can see that the line OE is a side of both triangles. When the same side is shared by two triangles, we say that the side is common. So, we have a pair of equal sides.

This matches the 90°HS (90°, hypotenuse, side) case for congruency.

Now we can write the proof of the congruency:

In ΔOCE and ΔOBE:

  1. OE^C=OE^B=90° (given)
  2. OC=OB (radii)
  3. OE is common

ΔOCEΔOBE (90°HS)


STEP: Deduce the length of BE
[−3 points ⇒ 0 / 5 points left]

The triangles are congruent, so their matching sides and angles are congruent.

BE=CE(ΔOCEΔOBE)BE=3 units
NOTE: We only know that BE=CE because the triangles are congruent. So, we must write "ΔOCEΔOBE" as the reason for this statement.

Submit your answer as: andandandand

Deductions from congruency

In the diagram below, O is the centre of the circle, and B is the midpoint of ED (in other words, EB=DB). Also, DO^B=50°.

Determine the size of EO^B.

INSTRUCTION: There are often many different ways to answer geometry questions. You must answer this question by completing the proof below.
Answer:

In ΔOEB and ΔODB:

  1. EB=DB (given)
  2. OE=OD (radii)

ΔOEBΔODB

EOB=
EOB= °

numeric
HINT: <no title>
[−0 points ⇒ 5 / 5 points left]

You cannot determine the size of EO^B directly using the geometry facts that you know. The only option is to prove that the two triangles are congruent, and then match up the angles.


STEP: Prove congruency
[−2 points ⇒ 3 / 5 points left]

We cannot determine the size of EO^B directly using the geometry facts that we know. First we must prove that the two triangles are congruent, and then use this fact to match up the angles.

The four cases for congruency are SSS, SAA, SAS, and 90°HS. When we prove congruency, we must first decide which of the four cases we can prove before writing out the proof.

We were told that EB=DB, so we have a pair of equal sides.

We were also told that O is the centre of the circle. The distance from the centre of a circle to its circumference is always the same length. It is called the radius. We can use this fact to prove that OE=OD, so we have a pair of equal sides.

We can see that the line OB is a side of both triangles. When the same side is shared by two triangles, we say that the side is common. So, we have a pair of equal sides.

This matches the SSS (side, side, side) case for congruency.

Now we can write the proof of the congruency:

In ΔOEB and ΔODB:

  1. EB=DB (given)
  2. OE=OD (radii)
  3. OB is common

ΔOEBΔODB (SSS)


STEP: Deduce the size of EO^B
[−3 points ⇒ 0 / 5 points left]

The triangles are congruent, so their matching sides and angles are congruent.

EO^B=DO^B(ΔOEBΔODB)EO^B=50 °
NOTE: We only know that EO^B=DO^B because the triangles are congruent. So, we must write "ΔOEBΔODB" as the reason for this statement.

Submit your answer as: andandandand

Consequences of congruency with calculations

In this diagram, ΔJKLΔXZY.

Determine the value of q, giving reasons for your answers.

INSTRUCTION: There is more than one step to solve this problem. You should write your final answer for q. Then choose the option which contains all of the reasons that you needed to work it out.
Answer:

q= units

In my working out, I used the following reason(s):

numeric
HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

Since the triangles are congruent, their matching sides will be equal. Can you use a geometry reason to find the length of the side which matches q?


STEP: Match up the sides of the congruent triangles
[−1 point ⇒ 2 / 3 points left]

If two triangles are congruent, their matching angles and sides will be equal.

We were told that ΔJKLΔXZY which means that JK is the same length as XZ. In the same way, KL is the same length as ZY, and JL is the same length as XY. The matching angles are also equal, but for this question we only need to think about the sides.

This means that q=KL, because ZY is equivalent to KL. We use the reason (ΔJKLΔXZY) to remind ourselves that the sides are only equal because the triangles are congruent.


STEP: Find the value of the missing side
[−2 points ⇒ 0 / 3 points left]
NOTE: In your working out, the reasons you used must be written next to the line in which you first used the relevant geometry fact. You can't just write all of your reasons at the end. Make sure you read the solution to see where you should have written your reasons.

We don't have enough information to find the size of q in triangle XZY.

But we know that q=KL, and we have enough information to find KL in triangle JKL, using the theorem of Pythagoras.

KL is the hypotenuse, because it is opposite the 90° angle.
(KL)2=(27)2+(36)2(Pythagoras)(KL)2=2025KL=45q=45(ΔJKLΔXZY)

Submit your answer as: and

Consequences of congruency with calculations

In this diagram, ΔACBΔQRP.

Determine the value of k, giving reasons for your answers.

INSTRUCTION: There is more than one step to solve this problem. You should write your final answer for k. Then choose the option which contains all of the reasons that you needed to work it out.
Answer:

k= °

In my working out, I used the following reason(s):

numeric
HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

Since the triangles are congruent, their matching angles will be equal. Can you use a geometry reason to find the size of the angle which matches k?


STEP: Match up the angles of the congruent triangles
[−1 point ⇒ 2 / 3 points left]

If two triangles are congruent, their matching angles and sides will be equal.

We were told that ΔACBΔQRP which means that A^ is the same size as Q^. In the same way, C^ is the same size as R^, and B^ is the same size as P^. The matching sides are also equal, but for this question we only need to think about the angles.

This means that k=A^, because Q matches A. We use the reason (ΔACBΔQRP) to remind ourselves that the angles are only equal because the triangles are congruent.


STEP: Find the value of the missing angle
[−2 points ⇒ 0 / 3 points left]
NOTE: In your working out, the reasons you used must be written next to the line in which you first used the relevant geometry fact. You can't just write all of your reasons at the end. Make sure you read the solution to see where you should have written your reasons.

We don't have enough information to find the size of k in triangle QRP.

But we know that k=A^ and we have enough information to find A^ in triangle ACB, using sum of angles in a triangle.

41°+61°+A^=180°(sum of s in Δ)A^=78°k=78°(ΔACBΔQRP)

Submit your answer as: and

Consequences of congruency with calculations

In this diagram, ΔBACΔKLJ.

Determine the value of z, giving reasons for your answers.

INSTRUCTION: There is more than one step to solve this problem. You should write your final answer for z. Then choose the option which contains all of the reasons that you needed to work it out.
Answer:

z= °

In my working out, I used the following reason(s):

numeric
HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

Since the triangles are congruent, their matching angles will be equal. Can you use a geometry reason to find the size of the angle which matches z?


STEP: Match up the angles of the congruent triangles
[−1 point ⇒ 2 / 3 points left]

If two triangles are congruent, their matching angles and sides will be equal.

We were told that ΔBACΔKLJ which means that B^ is the same size as K^. In the same way, A^ is the same size as L^, and C^ is the same size as J^. The matching sides are also equal, but for this question we only need to think about the angles.

This means that z=B^, because K matches B. We use the reason (ΔBACΔKLJ) to remind ourselves that the angles are only equal because the triangles are congruent.


STEP: Find the value of the missing angle
[−2 points ⇒ 0 / 3 points left]
NOTE: In your working out, the reasons you used must be written next to the line in which you first used the relevant geometry fact. You can't just write all of your reasons at the end. Make sure you read the solution to see where you should have written your reasons.

We don't have enough information to find the size of z in triangle KLJ.

But we know that z=B^ and we have enough information to find B^ in triangle BAC, using sum of angles in a triangle.

40°+98°+B^=180°(sum of s in Δ)B^=42°z=42°(ΔBACΔKLJ)

Submit your answer as: and

Identify congruency in overlapping triangles

Consider the diagram below:

VW and YX are both perpendicular to WX. Also, V^=x, Y^=x, and VZ=ZY.

Identify which triangle is congruent to ΔYXW. Give a reason for the congruency.

INSTRUCTIONS:
  • A full congruency proof is not required for this question.
  • Use upper case letters only.
Answer: ΔYXWΔ
string
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

The triangle you've been given has sides and angles which are equal to sides and angles in another triangle. Use one of the cases for congruency (SSS, SAA, SAS and 90°HS) to match up the triangles.


STEP: Match up equal sides and angles and select the case for congruency
[−2 points ⇒ 0 / 2 points left]

WX is a side of ΔYXW and ΔVWX. We say that WX is common.

We were given VW^X=YX^W=90°.

We were also given that WV^X=x=WY^X.

This means that ΔYXWΔVWX(SAA).

TIP: You must name ΔVWX in that exact order, because that is the order in which the equal angles and sides will match up with ΔYXW.

Submit your answer as: and

Identify congruency in overlapping triangles

Consider the diagram below:

In this diagram, PQ=RS=8 units, PS=QR=4 units, PT=TR=6 units, and TS=TQ=2 units.

Identify which triangle is congruent to ΔRQP. Give a reason for the congruency.

INSTRUCTIONS:
  • A full congruency proof is not required for this question.
  • Use upper case letters only.
Answer: ΔRQPΔ
string
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

The triangle you've been given has sides and angles which are equal to sides and angles in another triangle. Use one of the cases for congruency (SSS, SAA, SAS and 90°HS) to match up the triangles.


STEP: Match up equal sides and angles and select the case for congruency
[−2 points ⇒ 0 / 2 points left]

We were given PS=QR and PQ=RS.

PR is a side of ΔRQP and ΔPSR. We say that PR is common.

This means that ΔRQPΔPSR(SSS).

TIP: You must name ΔPSR in that exact order, because that is the order in which the equal angles and sides will match up with ΔRQP.

Submit your answer as: and

Identify congruency in overlapping triangles

Consider the diagram below:

JK and JL are equal in length. Also, KJ^N=JL^M and JN^K=JM^L.

Identify which triangle is congruent to ΔJKN. Give a reason for the congruency.

INSTRUCTIONS:
  • A full congruency proof is not required for this question.
  • Use upper case letters only.
Answer: ΔJKNΔ
string
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

The triangle you've been given has sides and angles which are equal to sides and angles in another triangle. Use one of the cases for congruency (SSS, SAA, SAS and 90°HS) to match up the triangles.


STEP: Match up equal sides and angles and select the case for congruency
[−2 points ⇒ 0 / 2 points left]

We were given JK=JL.

We were also given that KJ^N=JL^M and JN^K=JM^L.

This means that ΔJKNΔLJM(SAA).

TIP: You must name ΔLJM in that exact order, because that is the order in which the equal angles and sides will match up with ΔJKN.

Submit your answer as: and

Deductions from congruency: overlapping triangles

In the diagram below, QTRS. Also, PQ=PR and QP^T=PR^S.

  1. Complete the congruency proof below:

    Answer:

    In ΔPQT and ΔPRS:

    1. QPT=PRS(given)
    2. PQ=PR(given)

    ΔPTQΔ

    string
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    You are asked to prove congruency in ΔPQT and ΔPRS. How can you use the information given to prove that another pair of sides or angles are equal in these two triangles?


    STEP: Prove another pair of angles equal
    [−2 points ⇒ 2 / 4 points left]

    We have already proved a pair of matching sides and angles. We must prove another pair of sides or angles.

    In ΔPQT and ΔPRS:

    1. QP^T=PR^S(given)
    2. PQ=PR(given)
    3. RS^P=QT^P (corresp s;QTRS)

    STEP: Name the triangles correctly and give a reason for the congruency
    [−2 points ⇒ 0 / 4 points left]

    We have proved the SAA (side, angle, angle) congruency case. We must make sure that we label the triangles in the correct order.

    If we reflect and rotate the triangles, we can see the matching vertices more clearly:

    So, ΔPTQΔRSP (SAA).


    Submit your answer as: andandand
  2. You are now given that RS=6 and QT=8. Hence, determine the length of TS.

    Answer: TS= units
    numeric
    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    TS is the difference between the length of PS and PT. Can you use congruency to match these sides with sides that you know the length of?


    STEP: Use congruency to determine PS and PT
    [−2 points ⇒ 1 / 3 points left]

    PS is made out of PT and TS put together. So PS=PT+TS. If we know PS and PT, we can calculate TS.

    RS=6(given)PT=RS=6(ΔPTQΔRSP)
    QT=8(given)PS=QT=8(ΔPTQΔRSP)

    STEP: Hence, determine TS
    [−1 point ⇒ 0 / 3 points left]
    PS=PT+TS8=6+TSTS=2 units
    NOTE: Although you did not need to give reasons for this question, it is very important to include them in your written out answers. This is so that someone else can understand your thinking. Read our solution carefully to see where each reason should be!

    Submit your answer as:

Deductions from congruency: overlapping triangles

In the diagram below, BECD. Also, AB=AC and BA^E=AC^D.

  1. Complete the congruency proof below:

    Answer:

    In ΔABE and ΔACD:

    1. BAE=ACD(given)
    2. AB=AC(given)

    ΔBEAΔ

    string
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    You are asked to prove congruency in ΔABE and ΔACD. How can you use the information given to prove that another pair of sides or angles are equal in these two triangles?


    STEP: Prove another pair of angles equal
    [−2 points ⇒ 2 / 4 points left]

    We have already proved a pair of matching sides and angles. We must prove another pair of sides or angles.

    In ΔABE and ΔACD:

    1. BA^E=AC^D(given)
    2. AB=AC(given)
    3. CD^A=BE^A (corresp s;BECD)

    STEP: Name the triangles correctly and give a reason for the congruency
    [−2 points ⇒ 0 / 4 points left]

    We have proved the SAA (side, angle, angle) congruency case. We must make sure that we label the triangles in the correct order.

    If we reflect and rotate the triangles, we can see the matching vertices more clearly:

    So, ΔBEAΔADC (SAA).


    Submit your answer as: andandand
  2. You are now given that CD=5 and BE=6. Hence, determine the length of ED.

    Answer: ED= units
    numeric
    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    ED is the difference between the length of AD and AE. Can you use congruency to match these sides with sides that you know the length of?


    STEP: Use congruency to determine AD and AE
    [−2 points ⇒ 1 / 3 points left]

    AD is made out of AE and ED put together. So AD=AE+ED. If we know AD and AE, we can calculate ED.

    CD=5(given)AE=CD=5(ΔBEAΔADC)
    BE=6(given)AD=BE=6(ΔBEAΔADC)

    STEP: Hence, determine ED
    [−1 point ⇒ 0 / 3 points left]
    AD=AE+ED6=5+EDED=1 units
    NOTE: Although you did not need to give reasons for this question, it is very important to include them in your written out answers. This is so that someone else can understand your thinking. Read our solution carefully to see where each reason should be!

    Submit your answer as:

Deductions from congruency: overlapping triangles

In the diagram below, BECD. Also, AB=AC and BA^E=AC^D.

  1. Complete the congruency proof below:

    Answer:

    In ΔABE and ΔACD:

    1. BAE=ACD(given)
    2. AB=AC(given)

    ΔBAEΔ

    string
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    You are asked to prove congruency in ΔABE and ΔACD. How can you use the information given to prove that another pair of sides or angles are equal in these two triangles?


    STEP: Prove another pair of angles equal
    [−2 points ⇒ 2 / 4 points left]

    We have already proved a pair of matching sides and angles. We must prove another pair of sides or angles.

    In ΔABE and ΔACD:

    1. BA^E=AC^D(given)
    2. AB=AC(given)
    3. CD^A=BE^A (corresp s;BECD)

    STEP: Name the triangles correctly and give a reason for the congruency
    [−2 points ⇒ 0 / 4 points left]

    We have proved the SAA (side, angle, angle) congruency case. We must make sure that we label the triangles in the correct order.

    If we reflect and rotate the triangles, we can see the matching vertices more clearly:

    So, ΔBAEΔACD (SAA).


    Submit your answer as: andandand
  2. You are now given that CD=18 and BE=22. Hence, determine the length of ED.

    Answer: ED= units
    numeric
    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    ED is the difference between the length of AD and AE. Can you use congruency to match these sides with sides that you know the length of?


    STEP: Use congruency to determine AD and AE
    [−2 points ⇒ 1 / 3 points left]

    AD is made out of AE and ED put together. So AD=AE+ED. If we know AD and AE, we can calculate ED.

    CD=18(given)AE=CD=18(ΔBAEΔACD)
    BE=22(given)AD=BE=22(ΔBAEΔACD)

    STEP: Hence, determine ED
    [−1 point ⇒ 0 / 3 points left]
    AD=AE+ED22=18+EDED=4 units
    NOTE: Although you did not need to give reasons for this question, it is very important to include them in your written out answers. This is so that someone else can understand your thinking. Read our solution carefully to see where each reason should be!

    Submit your answer as:

Deductions from congruency: evaluating proofs

In the diagram below, DBCA. Also, CB=13, BD=12, and DA=5.

Prove that CB^D=AB^D.

Talwar has already answered the question: his proof is written below. Look carefully at his proof and identify where he has made his mistake.

CB^D=AB^D(ΔCDBΔADB)
Answer:

Talwar's proof is .

Talwar must:

HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

What must you do before assuming that two triangles are congruent?


STEP: Identify the error in the proof
[−3 points ⇒ 0 / 3 points left]

We cannot use congruency to deduce that CB^D=AB^D unless we are certain that the triangles are congruent. Sometimes, we are told in the question that the triangles are congruent. However, this is not the case here. We must first prove that ΔCDBΔADB before using this fact to answer the question.

Here is the completed proof:

In ΔCDB and ΔADB:

  1. CD^B=AD^B=90° (given)
  2. BA2=122+52 (Pythagoras)
    BA=13
    BA=CB
  3. BD is common

ΔCDBΔADB (90°HS)

CB^D=AB^D (ΔCDBΔADB)


Submit your answer as: and

Deductions from congruency: evaluating proofs

In the diagram below, SPQR. Also, QP=26, PS=24, and SR=10.

Prove that QP^S=RP^S.

Lulamile has already answered the question: his proof is written below. But, he has made a mistake! Look carefully at his proof and identify where he has made his mistake.

Line
one In ΔQSP and ΔRSP:
two 1. QS^P=RS^P=90° (given)
three 2. PR2=242+102 (Pythagoras)
four PR=26
five PR=QP
six 3. PS is common
seven ΔQSPΔRSP (90°HS)
eight QP^S=RP^S
Answer:

The mistake is on line .

Replace this line with:

HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

Read through the proof carefully, following every step. Is anything incorrect? Is there anything missing?


STEP: Identify the error in the proof
[−3 points ⇒ 0 / 3 points left]

This option does not have a reason for the deduction on the last line. We know that the angles are equal only because ΔQSPΔRSP. So, we must include this on the last line.

Line
one In ΔQSP and ΔRSP:
two 1. QS^P=RS^P=90° (given)
three 2. PR2=242+102 (Pythagoras)
four PR=26
five PR=QP
six 3. PS is common
seven ΔQSPΔRSP (90°HS)
eight QP^S=RP^S (ΔQSPΔRSP)

Submit your answer as: and

Deductions from congruency: evaluating proofs

In the diagram below, RSQP. Also, QS=15, SR=12, and RP=9.

Prove that QS^R=PS^R.

Tlotliso has already answered the question: her proof is written below. But, she has made a mistake! Look carefully at her proof and identify where she has made her mistake.

Line
one In ΔQRS and ΔPRS:
two 1. SR is common
three 2. SP2=122+92 (Pythagoras)
four SP=15
five SP=QS
six 3. QR^S=PR^S=90° (given)
seven ΔQRSΔPRS
eight QS^R=PS^R (ΔQRSΔPRS)
Answer:

The mistake is on line .

Replace this line with:

HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

Read through the proof carefully, following every step. Is anything incorrect? Is there anything missing?


STEP: Identify the error in the proof
[−3 points ⇒ 0 / 3 points left]

This option does not have a reason for the congruency on line seven. We must always write the reason why we know that the triangles are congruent: in this case it is 90°HS.

Line
one In ΔQRS and ΔPRS:
two 1. SR is common
three 2. SP2=122+92 (Pythagoras)
four SP=15
five SP=QS
six 3. QR^S=PR^S=90° (given)
seven ΔQRSΔPRS (90°HS)
eight QS^R=PS^R (ΔQRSΔPRS)

Submit your answer as: and

Deductions from congruency: overlapping triangles

In the diagram below, TR=TQ and PQ^R=SR^Q=90°.

  1. Khethang needs to prove that ΔPQRΔSRQ. He has already started and his incomplete proof is written below:

    In ΔPQR and ΔSRQ:

    1. PQ^R=SR^Q=90° (given)
    2. QR is common
      ...
    Answer:

    How should Khethang complete his proof? Choose the best option.

    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    You are asked to prove that ΔPQRΔSRQ. How can you use the information given to prove that another pair of sides or angles are equal in these two triangles?


    STEP: Complete the proof
    [−3 points ⇒ 0 / 3 points left]

    We need to use the given information to prove that another pair of sides or angles is equal, and hence prove a case for congruency.

    A ✘

    3. TQ=TR (given)
    ΔPQRΔSRQ (90°HS)

    TQ is equal to TR. But, these are not sides in the triangles that we are trying to prove are congruent. (They are only parts of the sides in those triangles, which is not enough.)
    B ✔

    3. TQ^R=TR^Q (s opp equal sides)
    ΔPQRΔSRQ (SAA)

    This option uses geometry reasons correctly to prove that TQ^R=TR^Q. This is a pair of matching angles in both triangles. So this proves that ΔPQRΔSRQ, using SAA.
    C ✘

    3. PT^Q=ST^R (vert opp s)
    ΔPQRΔSRQ (SAA)

    PT^Q is equal to ST^R, because the angles are vertically opposite. But, these are not angles in the triangles that we are trying to prove are congruent.
    D ✘

    3. PR is the hypotenuse of ΔPQR
    Also, SQ is the hypotenuse of ΔSRQ.
    PR=SQ (hypotenuse with common base)
    ΔPQRΔSRQ (90°HS)

    PR and SQ are hypotenuses of ΔPQR and ΔSRQ respectively. But, this does not mean that they have to be equal in length. Hypotenuse with common base does not refer to any of our geometry theorems. It is just a made up reason. So, this option does not prove that PR=SQ.

    The completed proof is shown below. The diagrams are also separated so that the congruency is easier to see.

    In ΔPQR and ΔSRQ:

    1. PQ^R=SR^Q=90° (given)
    2. QR is common
    3. TQ^R=TR^Q (s opp equal sides)

    ΔPQRΔSRQ (SAA)


    Submit your answer as:
  2. You are now given that QR=6 and RS=9.

    Hence, determine the length of PR.

    INSTRUCTION: Round off your answer to two decimal places.
    Answer: PR= units
    numeric
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    Since the triangles are congruent, their matching sides will be equal. Can you use geometry (with reasons) to find the length of the side which matches PR? Remember that there are right-angled triangles in this diagram.


    STEP: Use the theorem of Pythagoras to calculate SQ
    [−3 points ⇒ 1 / 4 points left]

    We do not have enough information to calculate PR directly. But, in ΔSRQ we can calculate SQ using the theorem of Pythagoras. Then we will use the fact that PR and SQ are matching sides in congruent triangles, to find PR.

    SQ2=SR2+QR2(Pythagoras)SQ2=92+62SQ2=117SQ=117=10,81665...10,82 units

    STEP: Hence, determine PR using congruency
    [−1 point ⇒ 0 / 4 points left]

    We have already proven that the triangles are congruent, so we know that their matching sides and angles are equal.

    PR=SQ(ΔPQRΔSRQ)PR=10,82 units
    NOTE: Although you did not need to give reasons for this question, it is very important to include them in your written out answers. This is so that someone else can understand your thinking. Read our solution carefully to see where each reason should be!

    Submit your answer as:

Deductions from congruency: overlapping triangles

In the diagram below, TR=TQ and PQ^R=SR^Q=90°.

  1. Adekemi needs to prove that ΔPQRΔSRQ. She has already started and her incomplete proof is written below:

    In ΔPQR and ΔSRQ:

    1. PQ^R=SR^Q=90° (given)
    2. QR is common
      ...
    Answer:

    How should Adekemi complete her proof? Choose the best option.

    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    You are asked to prove that ΔPQRΔSRQ. How can you use the information given to prove that another pair of sides or angles are equal in these two triangles?


    STEP: Complete the proof
    [−3 points ⇒ 0 / 3 points left]

    We need to use the given information to prove that another pair of sides or angles is equal, and hence prove a case for congruency.

    A ✘

    3. PT^Q=ST^R (vert opp s)
    ΔPQRΔSRQ (SAA)

    PT^Q is equal to ST^R, because the angles are vertically opposite. But, these are not angles in the triangles that we are trying to prove are congruent.
    B ✘

    3. TQ=TR (given)
    ΔPQRΔSRQ (90°HS)

    TQ is equal to TR. But, these are not sides in the triangles that we are trying to prove are congruent. (They are only parts of the sides in those triangles, which is not enough.)
    C ✘

    3. PR is the hypotenuse of ΔPQR
    Also, SQ is the hypotenuse of ΔSRQ.
    PR=SQ (hypotenuse with common base)
    ΔPQRΔSRQ (90°HS)

    PR and SQ are hypotenuses of ΔPQR and ΔSRQ respectively. But, this does not mean that they have to be equal in length. Hypotenuse with common base does not refer to any of our geometry theorems. It is just a made up reason. So, this option does not prove that PR=SQ.
    D ✔

    3. TQ^R=TR^Q (s opp equal sides)
    ΔPQRΔSRQ (SAA)

    This option uses geometry reasons correctly to prove that TQ^R=TR^Q. This is a pair of matching angles in both triangles. So this proves that ΔPQRΔSRQ, using SAA.

    The completed proof is shown below. The diagrams are also separated so that the congruency is easier to see.

    In ΔPQR and ΔSRQ:

    1. PQ^R=SR^Q=90° (given)
    2. QR is common
    3. TQ^R=TR^Q (s opp equal sides)

    ΔPQRΔSRQ (SAA)


    Submit your answer as:
  2. You are now given that QR=8 and RS=12.

    Hence, determine the length of PR.

    INSTRUCTION: Round off your answer to two decimal places.
    Answer: PR= units
    numeric
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    Since the triangles are congruent, their matching sides will be equal. Can you use geometry (with reasons) to find the length of the side which matches PR? Remember that there are right-angled triangles in this diagram.


    STEP: Use the theorem of Pythagoras to calculate SQ
    [−3 points ⇒ 1 / 4 points left]

    We do not have enough information to calculate PR directly. But, in ΔSRQ we can calculate SQ using the theorem of Pythagoras. Then we will use the fact that PR and SQ are matching sides in congruent triangles, to find PR.

    SQ2=SR2+QR2(Pythagoras)SQ2=122+82SQ2=207SQ=207=14,42220...14,42 units

    STEP: Hence, determine PR using congruency
    [−1 point ⇒ 0 / 4 points left]

    We have already proven that the triangles are congruent, so we know that their matching sides and angles are equal.

    PR=SQ(ΔPQRΔSRQ)PR=14,42 units
    NOTE: Although you did not need to give reasons for this question, it is very important to include them in your written out answers. This is so that someone else can understand your thinking. Read our solution carefully to see where each reason should be!

    Submit your answer as:

Deductions from congruency: overlapping triangles

In the diagram below, EC=EB and AB^C=DC^B=90°.

  1. Ismail needs to prove that ΔABCΔDCB. He has already started and his incomplete proof is written below:

    In ΔABC and ΔDCB:

    1. AB^C=DC^B=90° (given)
    2. BC is common
      ...
    Answer:

    How should Ismail complete his proof? Choose the best option.

    HINT: <no title>
    [−0 points ⇒ 3 / 3 points left]

    You are asked to prove that ΔABCΔDCB. How can you use the information given to prove that another pair of sides or angles are equal in these two triangles?


    STEP: Complete the proof
    [−3 points ⇒ 0 / 3 points left]

    We need to use the given information to prove that another pair of sides or angles is equal, and hence prove a case for congruency.

    A ✘

    3. AE^B=DE^C (vert opp s)
    ΔABCΔDCB (SAA)

    AE^B is equal to DE^C, because the angles are vertically opposite. But, these are not angles in the triangles that we are trying to prove are congruent.
    B ✔

    3. EB^C=EC^B (s opp equal sides)
    ΔABCΔDCB (SAA)

    This option uses geometry reasons correctly to prove that EB^C=EC^B. This is a pair of matching angles in both triangles. So this proves that ΔABCΔDCB, using SAA.
    C ✘

    3. EB=EC (given)
    ΔABCΔDCB (90°HS)

    EB is equal to EC. But, these are not sides in the triangles that we are trying to prove are congruent. (They are only parts of the sides in those triangles, which is not enough.)
    D ✘

    3. AC is the hypotenuse of ΔABC
    Also, DB is the hypotenuse of ΔDCB.
    AC=DB (hypotenuse with common base)
    ΔABCΔDCB (90°HS)

    AC and DB are hypotenuses of ΔABC and ΔDCB respectively. But, this does not mean that they have to be equal in length. Hypotenuse with common base does not refer to any of our geometry theorems. It is just a made up reason. So, this option does not prove that AC=DB.

    The completed proof is shown below. The diagrams are also separated so that the congruency is easier to see.

    In ΔABC and ΔDCB:

    1. AB^C=DC^B=90° (given)
    2. BC is common
    3. EB^C=EC^B (s opp equal sides)

    ΔABCΔDCB (SAA)


    Submit your answer as:
  2. You are now given that BC=10 and DB=14.

    Hence, determine the length of AB.

    INSTRUCTION: Round off your answer to two decimal places.
    Answer: AB= units
    numeric
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    Since the triangles are congruent, their matching sides will be equal. Can you use geometry (with reasons) to find the length of the side which matches AB? Remember that there are right-angled triangles in this diagram.


    STEP: Use the theorem of Pythagoras to calculate DC
    [−3 points ⇒ 1 / 4 points left]

    We do not have enough information to calculate AB directly. But, in ΔDCB we do have enough information to calculate the length of DC. Since AB and DC are matching sides in congruent triangles, they will be equal.

    DB2=DC2+BC2(Pythagoras)142=DC2+102196=DC2+100196100=DC2+10010095=DC2DC2=95DC=95=9,79795...9,8 units

    STEP: Hence, determine AB using congruency
    [−1 point ⇒ 0 / 4 points left]

    We have already proven that the triangles are congruent, so we know that their matching sides and angles are equal.

    AB=DC(ΔABCΔDCB)AB=9,8 units
    NOTE: Although you did not need to give reasons for this question, it is very important to include them in your written out answers. This is so that someone else can understand your thinking. Read our solution carefully to see where each reason should be!

    Submit your answer as:

3. Properties of quadrilaterals

Apply diagonal properties of a rhombus

In the diagram below, PQRS is a rhombus. Also, PS=10 and PR=12.

Determine, with reasons, the length of SM.

INSTRUCTION: Answer this question by completing the steps below.
Answer: PM= units
Also, PM^S= °
Hence, SM= units
numeric
numeric
numeric
HINT: <no title>
[−0 points ⇒ 6 / 6 points left]

What do you need to know about a triangle, before you can use the theorem of Pythagoras? Use the properties that you know about the diagonals of a rhombus.


STEP: Decide on a strategy
[−1 point ⇒ 5 / 6 points left]

We are going to use the theorem of Pythagoras in ΔPMS to determine PM. To do this, we will use rhombus diagonal properties to determine the necessary sides, and to prove that ΔPMS is right-angled.


STEP: Apply diagonal properties to determine PM
[−1 point ⇒ 4 / 6 points left]
The diagonals of a rhombus bisect each other.

PR=12 (given)
And, PM=MR (diags of rhombus)
PM= 6 units


STEP: Show that ΔPMS is right-angled
[−2 points ⇒ 2 / 6 points left]
The diagonals of a rhombus intersect at 90°.

PM^S=90° (diags of rhombus)


STEP: Use the theorem of Pythagoras to determine the length of SM
[−2 points ⇒ 0 / 6 points left]

We have shown that ΔPMS is right-angled. Now, we can use the theorem of Pythagoras to work out SM.

SM2+PM2=PS2 (Pythagoras)

SM2+62=102SM2+36=100SM2=64SM=64SM=8 units

Submit your answer as: andandandandand

Reasoning about properties of quadrilaterals

Consider the following statement and decide if it is always true, sometimes true, or never true.

Rectangles have two pairs of opposite sides that are parallel.
Answer:

The statement is true.

HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Are there any quadrilaterals that make the statement true? Are there any quadrilaterals that make the statement false?


STEP: Think about whether any quadrilaterals make the statement true
[−2 points ⇒ 0 / 2 points left]

The opposite sides of a rectangle are always parallel, because a rectangle is a special type of parallelogram.

So, the statement is always true.


Submit your answer as:

Identify all diagonal properties

Think about the diagonals of a kite.

Select whether each of the following statements is true or false for all kites.

Answer:
  1. The diagonals bisect each other.
  2. Both diagonals are the same length.
  3. The diagonals intersect at 90°.
  4. The diagonals bisect the corner angles.
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

Draw a standard kite and compare the diagram to each diagonal property.


STEP: Identify the diagonal properties
[−4 points ⇒ 0 / 4 points left]

When we draw in the diagonals, we create several pairs of congruent triangles, including ΔPQTΔPST. (This is a bit harder to prove - but you should still try!) This means that T1 and T2 are equal. We also know that T1 and T2 add up to 180°. So, they must both be 90°. So, the diagonals intersect at 90°.

NOTE: One diagonal of a kite, QS, is bisected by the other. But PR is not bisected. So, we do not say that the diagonals bisect each other in a kite.

This is the only property that we can prove for certain about the diagonals of all kites. The following table shows the correct answers:

Diagonal property Definitely true for any kite?
The diagonals bisect each other. False
Both diagonals are the same length. False
The diagonals intersect at 90°. True
The diagonals bisect the corner angles. False

Submit your answer as: andandand

Congruent triangles

Have a look at the following triangles, which are drawn to scale:

  1. Answer the two questions below.

    Answer:

    Are these triangles congruent?
    If yes, select the reason. If no, select 'not congruent.'

    HINT: <no title>
    [−0 points ⇒ 2 / 2 points left]

    The word "congruent" means "same shape". Look carefully at the triangles and the given information. Can you prove that these triangles are congruent? If so, remember to provide a correct reason.


    STEP: Examine the picture to see if the triangles are identical
    [−2 points ⇒ 0 / 2 points left]

    The word "congruent" means "identical." If the triangles have exactly the same shape, then they are congruent.

    There are four different ways to determine if two triangles are congruent.

    1. Right angle, hypotenuse, side (RHS): the two triangles are right-angled triangles, the hypotenuse of one triangle is the same length as the hypotenuse of the other triangle and one of the other pairs of corresponding sides are equal in length.
    2. Side, side, side (SSS): all three pairs of corresponding sides in the two triangles have equal lengths.
    3. Side, angle, side (SAS): two pairs of corresponding sides and the included angle are equal.
    4. Angle, angle, side (AAS): two pairs of corresponding angles are equal and one pair of corresponding sides have equal lengths.

    We are given information about two angles and one side: there are two pairs of corresponding angles that are equal (indicated by the dots and the stars), and one pair of sides which are the same length.

    Therefore, these two triangles are congruent, and the reason is AAS.


    Submit your answer as: and
  2. Which of the following options correctly completes the statement below?

    Answer: ΔMPNΔ
    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]
    When shapes are congruent, you must label them in the order of their corresponding angles. "Corresponding" means "matching." Look at the triangles: does M^ correspond to angle Q^,R^, or S^?
    STEP: Compare the triangles to complete the statement
    [−1 point ⇒ 0 / 1 points left]

    When we name a pair of congruent triangles we need to pay careful attention to the order of the angles and which sides are equal: the triangles must be named according to their corresponding angles and sides. "Corresponding" means "matching."

    The first triangle can be named in any order - we are given ΔMPN. The second triangle needs to be named so that we can read off the corresponding angles and sides: we need to ensure that the corresponding sides and angles match up.

    For example, vertex M corresponds to vertex Q. Therefore, the answer must start with Q. Now think about which vertex corresponds to vertex P to figure out which letter should be second in the answer.

    The correct choice to complete the statement is ΔQSR.


    Submit your answer as:

Calculate angles between parallel lines

The diagram represents transversal line EF that intersects with the straight lines AB and CD respectively. AB CD.

The intersecting lines create different angles which are labeled for you. Carefully study the diagram and answer the questions that follow.

Answer:
  1. What is the value of x? °
  2. What is the value of y? °
  3. What is the value of z? °
numeric
numeric
numeric
HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

STEP: Determine the unknown angles
[−3 points ⇒ 0 / 3 points left]

If the two lines are parallel, the four angles around the first intersection are the same as the four angles around the second intersection. We can use this, and the fact that angles on a straight line add up to 180°, to determine the unknown angles.

Therefore:

  1. x=55°
  2. y=55°
  3. z=125°

Submit your answer as: andand

Proof: is the quadrilateral a parallelogram?

The figure below shows quadrilateral ABCD with ADBC and AD=BC. Diagonal BD is shown with a dashed line. A=y and C=54° ; ADB=79° and BDC=x.

  1. The steps and reasons below prove that ABCD is a parallelogram. But some steps and reasons are incomplete. Choose the correct steps and reasons from the drop down boxes to complete the proof.

    Answer:
    STEPS REASONS
    ADB=DBC Alternate interior angles are equal if ADBC
    BD=DB
    Given
    ΔABD=ΔCDB Side-Angle-Side for congruent triangles (SAS)
    ABD=CDB
    ABD & CDB are equal alternate interior angles
    ABCD is a parallelogram Definition of a parallelogram (opposite sides parallel)
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    We already know that side AD is parallel to side BC. The proof above shows that the other two sides are also parallel, based on the information given about the angles. Work your way down the proof, one step at a time, to choose the correct steps and reasons.


    STEP: Work your way down the proof, choosing the answers along the way
    [−4 points ⇒ 0 / 4 points left]

    The overall strategy of the proof in this question is to show that the sides AB and DC are parallel. If we can show that, we can conclude that the quadrilateral is a parallelogram because we already know that AD is parallel to BC. But it takes six steps to first prove that AB and DC are parallel.

    The proof starts by using the parallel sides and the diagonal. The parallel lines and the diagonal invite us to use transversal geometry: alternating interior angles, corresponding angles, cointerior angles, and so on. So we can show that two of the angles are equal. Then the proof proceeds to show (in step 4) that the triangles ABD and CDB are congruent (which means that the triangles are identical).

    The complete proof is as follows:

    Steps Reasons
    ADB=DBC Alternate interior angles are equal if ADBC
    BD=DB Common side (in triangles ABD & CDB)
    AD=BC Given
    ΔABD=ΔCDB Side-Angle-Side for congruent triangles (SAS)
    ABD=CDB Corresponding angles in congruent triangles
    ABDC ABD & CDB are equal alternate interior angles
    ABCD is a parallelogram Definition of a parallelogram (opposite sides parallel)

    Submit your answer as: andandand
  2. Determine the value of y. The diagram is repeated here, but now with both sides labelled as parallel (which we know from Question 1).

    Answer: y= °
    numeric
    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]

    Use the fact that opposite angles in a parallelogram are equal.


    STEP: Use the fact that opposite angles in a parallelogram are equal
    [−1 point ⇒ 0 / 1 points left]

    ABCD is a parallelogram. Opposite angles of a parallelogram are equal. So A is equal to C.

    Therefore, y=54°.


    Submit your answer as:
  3. Determine the value of x.

    Answer: x= °
    numeric
    HINT: <no title>
    [−0 points ⇒ 2 / 2 points left]

    Start by writing an equation for the angles in triangle CBD. The sum of those angles, which include x, must be 180°.


    STEP: Set up an equation and solve for x
    [−2 points ⇒ 0 / 2 points left]

    We can solve this problem using the fact that angle x is inside of a triangle CBD.

    The sum of angles in a triangle is 180°, so based on the figure we can write:

    C+CDB+CBD=180°

    From Question 2 we know that y=54° (this is labelled in the diagram above). And because of alternate interior angles we also know that CBD=79°. Putting all of this together:

    C+CDB+CBD=180°54°+x+79°=180°x=180°54°79°x=47°
    NOTE: You can also solve this question using the fact that the angles in a quadrliateral have a sum of 360°. Using the fact that opposite angles in a parallelogram are equal, the equation would be:
    54°+54°+2(79°+x)=360°
    If you solve this you will also get the answer x=47°.

    The measure of angle x is 47°.


    Submit your answer as:

Choose one correct diagonal property

ABCD below is a square.

Answer:

Which of the following statements is definitely true about the diagonals of ABCD?

HINT: <no title>
[−0 points ⇒ 2 / 2 points left]
  • Diagonals go from one corner of the square to the opposite corner.
  • Bisect means to cut into two equal parts.
  • Perpendicular means that the lines meet at 90°.

STEP: Identify the diagonal properties
[−2 points ⇒ 0 / 2 points left]

A square is a special type of rectangle and a special type of rhombus. So it will have all of the diagonal properties of rectangles and of rhombuses.

The following table sums up all of the diagonal properties of a square:

Diagonal property Definitely true for any square?
The diagonals bisect each other.
Both diagonals are the same length.
The diagonals intersect at 90°.
The diagonals bisect the corner angles.

Of the options given, only A is definitely true for all squares.


Submit your answer as:

Apply diagonal properties

In the diagram below, PQ^T=24° and PS^T=99°.

Answer the following questions:

  1. What type of quadrilateral is PQRS? (Give the most specific name.)
  2. Determine the size of TS^R.
INSTRUCTION: You do not need to give any reasons for your answers in this question.
Answer:
  1. PQRS is a .
  2. TS^R= °
numeric
HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

Think about the diagonal properties of PQRS. How can these help you to determine TS^R?


STEP: Identify PQRS
[−1 point ⇒ 2 / 3 points left]

The most specific name for PQRS is a parallelogram. Even though PQRS is a type of trapezium (for example), this is not the most specific name for PQRS.

TIP: When asked to give the name of a quadrilateral, always give the most specific name.

STEP: Apply diagonal properties of a parallelogram to determine TS^R
[−2 points ⇒ 0 / 3 points left]

Remember the diagonal properties of a parallelogram:

Diagonal property Definitely true for a parallelogram?
The diagonals bisect each other.
Both diagonals are the same length.
The diagonals intersect at 90°.
The diagonals bisect the corner angles.

The diagonals of a parallelogram do not necessarily bisect the corner angles, so we cannot assume that TS^R is equal to PS^T. In fact, we must use the fact that alternate angles on parallel lines are equal to deduce that TS^R = PQ^T = 24°.



Submit your answer as: and

Congruent triangles

In the diagram below, ΔMNPΔMQP. Also, MPQN while NP=5 and MP=4.

  1. Calculate the value of x.
  2. Determine the length of QN.
Answer:
  1. x= units
  2. QN= units
numeric
numeric
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

The triangles are congruent. That means they are identical. Start by using this fact to label all the sides which are not already labelled.


STEP: Label everything you can in the figure
[−1 point ⇒ 3 / 4 points left]

The first question asks us to determine the length of the side of the diagram labelled x. The x is attached to side MQ.

Based on the labels in the diagram, we do not know enough to calculate x immediately. However, the question states that the two triangles are congruent. That means they are identical. We can use that information to label all the sides which are not already labelled. The question also states that MPQN, which means that there are two right angles. We should label those right angles too.

Now we can see two right-angled triangles. And we can see which sides are equal to each other. Both of the triangles include a side labelled x.


STEP: Use the theorem of Pythagoras to find the value of x
[−2 points ⇒ 1 / 4 points left]

Now calculate x using the theorem of Pythagoras. We will focus on triangle MQP, but you can use either one, because they are congruent.

In ΔMQP:QP=5(ΔMNPΔMQP)(5)2=x2+(4)2(Pythagoras)25=x2+162516=x2±9=x3=x

The length of MQ is 3 units.


STEP: Find the length of QN
[−1 point ⇒ 0 / 4 points left]

For the second question, we need to find the length of segment QN. This segment is twice as long as x, which we calculated above. So QN=2(3)=6 units.

The correct answers are:

  1. The length of side x is 3 units.
  2. The length of QN is 6 units.

Submit your answer as: and

Definitions of quadrilaterals

  1. Select the most correct definition for a rectangle.

    Answer: A rectangle is a:
    HINT: <no title>
    [−0 points ⇒ 2 / 2 points left]

    You need to choose the definition that lists all the properties which make the shape what it is. None of the essential properties should be missing from your definition. You also must not include any unnecessary properties.


    STEP: Choose the definition with the exact properties
    [−2 points ⇒ 0 / 2 points left]

    We use the properties of shapes to define them. We need to choose the definition that lists all the properties which make the shape what it is. None of the essential properties should be missing from the definition. Also, the definition must not include any unnecessary properties.

    • A quadrilateral with two pairs of opposite sides parallel, adjacent sides equal, and all corners 90° is a square.
    • A quadrilateral with two pairs of opposite sides parallel and adjacent sides equal is a rhombus.
    • A rectangle is a quadrilateral with two pairs of opposite sides parallel and all corners 90°.
    • A quadrilateral with at least one pair of opposite sides parallel is a trapezium.

    This is a diagram of a rectangle with the definition properties drawn in:

    NOTE: You can also define a rectangle as a quadrilateral with two pairs of opposite sides equal and all corner angles 90°.

    Submit your answer as:
  2. Complete the statement below:

    Answer:

    A has all of the same properties as a rectangle (plus some more), so it is a special type of rectangle.

    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]

    Look at the definition for a rectangle, and ask yourself which quadrilateral on the list has these same properties. It can have extra properties, but it cannot have less.


    STEP: Choose the quadrilateral that has at least the same properties as a rectangle
    [−1 point ⇒ 0 / 1 points left]

    A square has two pairs of opposite sides parallel and all corners 90°. This means that a square meets the definition of a rectangle, so we say that it is a special type of rectangle.

    NOTE: It does not matter that a square has extra properties: what matters is that it has enough properties to be called a rectangle.

    Submit your answer as:

Opposites of a parallelogram

Answer the following questions about the parallelogram ABCD. The parallelogram has the following sides and angles:

A^=y, B^=x, C^=60°, and D^=120°. The sides are AB=7, BC=w, CD=7, and AD=8.

Answer:
  1. What is the size of x? °
  2. What is the size of y? °
  3. What is the length of side w? units
numeric
numeric
numeric
HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

You do not need to calculate anything. You can read all three answers from the diagram. You need to use the properties of sides and angles for parallelograms.


STEP: Find the unknown angles and the length of the missing side
[−3 points ⇒ 0 / 3 points left]

We can answer this question based on the properties of parallelograms.

There are two properties of parallelograms which are useful in this question:

  • Opposite angles in a parallelogram are equal
  • Opposite sides of parallelograms have equal lengths

Therefore:

  • x=120°
  • y=60°
  • w=8units

Submit your answer as: andand

Identifying quadrilaterals

Which of the following are squares?

A B
C D
Answer: The squares are .
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Which of these quadrilaterals have the properties which meet the definition of a squares? They could have extra properties as well - that does not mean they are not squares.


STEP: Identify the quadrilaterals that meet the definition of a square
[−2 points ⇒ 0 / 2 points left]
A square is a quadrilateral with two pairs of opposite sides that are parallel, equal adjacent sides and corners that are 90°.

We can easily recognise Quadrilateral A as a square. This is the only square in the options, because none of the other quadrilaterals have all of the properties of a square.

NOTE: Sometimes a rhombus is referred to as a "tilted square". But, it is not a type of square because it does not have 90° corners, which is part of the definition of a square.

Submit your answer as:

Angles on a straight line

Line ST represents angles on one side of a straight line. a = 44° , b = x and c = 70°.

Answer the following questions about the diagram:

  1. What is the value of x?
  2. What type of angle is represented by x?
Answer:
  1. x= °
  2. The angle is .
numeric
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

It can be shown that the measurement of a straight angle is 180° or π radians.


STEP: Use the fact that angles on a straight line add to 180°
[−2 points ⇒ 0 / 2 points left]

Flat or straight angles are formed when the legs are pointing in exactly opposite directions. The two legs then form a single straight line through the vertex of the angle. Angles in a straight line add up to 180°.

a+b+c=180°44°+x+70°=180°x=66°

Submit your answer as: and

Angles on a straight line

Examine the diagram below, and answer the questions that follow.

  1. Calculate x.
  2. What type of angle is represented with x?
Answer:
  1. x= °
  2. The angle is: .
numeric
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

The angles must have a sum of 180°. Write an equation based on this information and solve for x.


STEP: Write an equation to summarise the angles
[−1 point ⇒ 1 / 2 points left]

The three angles must have a sum of 180° because ST is a straight line. We can write an equation based on this information:

30°+x+50°=180°(s on a str line)

STEP: Solve the equation and deterimine the type of the angle
[−1 point ⇒ 0 / 2 points left]

Now we can solve the equation. Once we have the answer we can determine the type of angle.

30°+x+50°=180°x=100°

The complete diagram, with all three angles known, is:

The missing angle is 100°, which is obtuse.


Submit your answer as: and

Prove the quadrilateral is a parallelogram

The figure below shows quadrilateral XWVU, which is a parallelogram. The 4 dashed red lines bisect each vertex (angle) of the parallelogram (angles X,W,V and U). So X1=X2, W1=W2, and so on. These dashed lines form quadrilateral EFGH inside of parallelogram XWVU.

Note the diagram is drawn to scale.

The table below shows a proof that quadrilaterial EFGH is a parallelogram. But 4 steps and 4 reasons are missing from the proof. Complete the proof by choosing the correct steps and reasons from each of the drop down boxes.

Answer:
STEPS REASONS
X=V Opposite angles in a parallelogram are equal
X2=V1 and X1=V2 Angles created by bisectors of equal angles are equal
Angles created by bisectors of equal angles are equal
XU=WV Opposite sides of a parallelogram are equal
ΔXUEΔVGW Congruent triangles (Angle-Side-Angle)
In quadrilateral EFGH:E2=G2 Congruent triangles have equal angles (ΔXUEΔVGW)
XW=UV Opposite sides of a parallelogram are equal
H1=F1
H1=H2 and F1=F2
Opposite angles in a parallelogram are equal
HINT: <no title>
[−0 points ⇒ 8 / 8 points left]

You should start at the top of the proof and work your way down. Try to understand each step and each reason, because you will need to know how the proof is developing from the top to the bottom.


STEP: Work down the proof and select the steps and reasons along the way
[−8 points ⇒ 0 / 8 points left]

The proof is about the quadrilateral in the middle of the figure, EFGH. The steps of the proof need to show that this quadrilateral is a parallelogram. We need to complete the proof by selecting the correct steps and reasons.

The question states that quadrilateral XWVU is a parallelogram. We know lots of facts about parallelograms. The question also states that the red dashed lines bisect the angles at each vertex. (This means the line splits the angle into two equal parts.) It is a good idea to mark everything we know on the diagram itself. The diagram below shows a number of new labels for equal angles and equal sides.

The strategy used in this proof can be summarised as follows: we want to prove that EFGH is a parallelogram. We can do that if we can show that the opposite angles inside EFGH are equal. So the proof shows that there are congruent triangles in the diagram. That leads to congruent angles. And from those congruent angles we can conclude that EFGH has two pairs of opposite angles which are equal.

NOTE: There are other ways to prove that quadrilateral EFGH is a parallelogram: this is not the only way to do it. On a test or exam, you might use a different set of steps and reasons. But for this question, you must find the steps and reasons which fit within the proof as it is presented.

The table below shows the correct answer choices (in the green blocks).

STEPS REASONS
X=V Opposite angles in a parallelogram are equal
X2=V1 and X1=V2 Angles created by bisectors of equal angles are equal
W=U Opposite angles in a parallelogram are equal
W2=U1 and W1=U2 Angles created by bisectors of equal angles are equal
XU=WV Opposite sides of a parallelogram are equal
ΔXUEΔVGW Congruent triangles (Angle-Side-Angle)
In quadrilateral EFGH:E2=G2 Congruent triangles have equal angles (ΔXUEΔVGW)
XW=UV Opposite sides of a parallelogram are equal
ΔXHWΔVFU Congruent triangles (Angle-Side-Angle)
H1=F1 Congruent triangles have equal angles (ΔXHWΔVFU)
H1=H2 and F1=F2 Vertically opposite angles are equal
EFGH is a parallelogram Opposite angles in a parallelogram are equal

Submit your answer as: andandandandandandand

Angles in a full turn

In the diagram below, line ST is a straight line with angles labelled above and below it.

Answer the following questions about the diagram:

Answer:
  1. x= °
  2. y= °
  3. y is .
numeric
numeric
HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

Angles on a straight line add up to 180°.


STEP: <no title>
[−3 points ⇒ 0 / 3 points left]

Line ST is a straight line, and angles on a straight line up to 180°. This means that all of the angles above ST add up to 180°:

30°+x+72°=180°(s on a str line)x=78°

In the same way, all of the angles below ST add up to 180°:

25°+55°+22°+y=180°(s on a str line)y=78°

Here is the completed diagram, with all of the angle values labelled:

The angles in a full turn (revolution) must add up to 360°. This is a good way to check our answers:

30°+78°+72°+25°+55°+22°+78°=360°

y=78°

y is an acute angle (less than 90°).


Submit your answer as: andand

Relationships between quadrilateral definitions

It is useful to think about quadrilaterals as a connected family. One type of quadrilateral can have all the same properties as another type of quadrilateral, plus some extra properties.

Select from the options to complete the statement about a parallelogram.

Answer: A parallelogram is a trapezium with:
HINT: <no title>
[−0 points ⇒ 1 / 1 points left]

What is the definition of a trapezium? What additional properties does a parallelogram have?


STEP: Compare the definitions for a parallelogram and a trapezium
[−1 point ⇒ 0 / 1 points left]
A trapezium is a quadrilateral with at least one pair of opposite sides parallel.
A parallelogram is a quadrilateral with two pairs of opposite sides parallel.

The following diagram shows a trapezium next to a parallelogram, with their defining properties filled in.

We can see that the parallelogram has the same properties as the trapezium, plus one additional pair of parallel sides.


Submit your answer as:

Using diagonals to prove properties

In the diagram below, JKLM is a quadrilateral with JK=KL and JKN=LKN=31°.

  1. How should we prove that JM^N=LM^N?

    Answer:
    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]

    You must only use the information you have been given. Can you use this information to prove that any triangles are congruent? Then, their matching angles and sides will be equal.


    STEP: Choose which triangles to prove congruent
    [−1 point ⇒ 0 / 1 points left]

    JM^N is a angle in ΔJKM and LM^N is the matching angle in ΔLKM. So, if we prove that ΔJKMΔLKM , then JM^N and LM^N must be equal because congruent triangles are equal in every way.

    We also have to check that we have enough information to prove that ΔJKMΔLKM . See if you can spot two pairs of sides and one pair of angles that are equal between the two triangles!


    Submit your answer as:
  2. The diagram of quadrilateral JKLM is repeated for convenience:

    Now, prove that JM^N=LM^N by selecting the correct options.

    Answer:

    In :

    1. is common.
    2. JKN=LKN (given)

    JM^N=LM^N

    HINT: <no title>
    [−0 points ⇒ 6 / 6 points left]

    Prove congruency by matching up pairs of sides and angles. Use the given information and geometry facts.


    STEP: Complete the proof
    [−6 points ⇒ 0 / 6 points left]
    TIP:
    • Look for any sides or angles that the two triangles have in common.
    • Then, write down any pairs of equal sides or angles that you have been given.
    • Finally, decide what else you need in order to prove that the two triangles are congruent, according to one of the four cases of congruency.

    The correct proof is:

    In ΔJKM and ΔLKM:

    1. KM is common.
    2. JK=KL (given)
    3. JKN=LKN (given)

    ΔJKMΔLKM (SAS)

    JM^N=LM^N (ΔJKMΔLKM)

    NOTE: You can extend this proof to prove that the diagonals of any rhombus bisect all of its vertices.

    Submit your answer as: andandandandand

Proof: is the quadrilateral a parallelogram?

The figure below shows quadrilateral QRST with Q=S=90° and R=T=90°.

The steps and reasons below prove that QRST is a parallelogram. However, there are two steps and two reasons missing. Choose the correct steps and reasons to complete the proof.

Answer:
STEPS REASONS
Q+T=180°
QRTS
Given (90°+90°=180°)
Co-interior angles of parallel lines have a sum of 180°
QRST is a parallelogram Definition of a parallelogram
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

You need to start at the top, and go down the proof line by line. Work out the answers as you go down the proof. And stay patient: this question requires a lot of careful thinking!


STEP: Consider the options and choose the correct steps and reasons
[−4 points ⇒ 0 / 4 points left]

The definition of a parallelogram is: "a quadrilateral with both pairs of opposite sides parallel." To prove that the quadrilateral is a parallelogram, we need to show that the opposite sides are parallel to each other. In other words, the proof should follow whatever steps are needed to show that side QR is parallel to side TS and that side RS is parallel to side QT.

This proof uses the angle values to show that adjacent angles must be co-interior angles within parallel lines. The table below explains the various steps of the proof, using colour to tie the explanation to the steps and reasons in the proof.

Explanations

The yellow rows in the table indicate the given information. This is information we can gather directly from the diagram. The size of all the angles are given.

The green rows represent the application of co-interior angles. Co-interior angles inside parallel lines are supplementary (they have a sum of 180°). Here we use this fact in reverse: if the angles are supplementary the lines are parallel.

Once we have shown that the opposite sides are parallel to each other, we can conclude that quadrilateral QRST is a parallelogram. Note that the final step of a proof will always be the fact that you are trying to prove.

Here is the completed proof with the correct steps and reasons.

Steps Reasons
Q+T=180° Given (90°+90°=180°)
QRTS Co-interior angles of parallel lines have a sum of 180°
T+S=180° Given (90°+90°=180°)
RSQT Co-interior angles of parallel lines have a sum of 180°
QRST is a parallelogram Definition of a parallelogram

Submit your answer as: andandand

Relationships between quadrilaterals: true or false

Consider the following statement and decide whether it is true or false.

All parallelograms are rectangles.
Answer: The statement is .
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Think about the definitions of the quadrilaterals and how they relate to each other. Does a parallelogram always have at least the same properties as a rectangle? Or, can you think of any parallelograms that are not types of rectangles?


STEP: Think about the definitions of the quadrilaterals
[−2 points ⇒ 0 / 2 points left]
A parallelogram is a quadrilateral with two pairs of opposite sides that are parallel.
A rectangle is a quadrilateral with two pairs of opposite sides that are parallel and corners that are 90°.

Consider the following parallelogram:

This shape meets the definition of a parallelogram. But, it does not meet the definition of a rectangle because corner angles are not 90°. So, we have come up with an example that disagrees with the statement. Therefore, the statement is false.

NOTE: An example that disagrees with the statement is called a counter-example. We only need one counter-example to prove that a statement is false. This is because if it is false for one example, then we cannot say that it is true for all - in other words, it is false.

Submit your answer as:

Proving diagonals using congruency

In the diagram below, WVXY is a quadrilateral with WY=VX and WV^X=VW^Y= 90°.

Prove that WX=VY.

  1. How should we prove that WX=VY?

    Answer:
    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]

    You must only use the information you have been given. Can you use this information to prove that any triangles are congruent? Then, their matching angles and sides will be equal.


    STEP: Choose which triangles to prove congruent
    [−1 point ⇒ 0 / 1 points left]

    The correct strategy is: Prove that ΔWVYΔVWX.

    WX is a side in ΔVWX and VY is the matching side in ΔWVY. So, if we prove that ΔWVYΔVWX, then WX and VY must be equal because congruent triangles are equal in every way.

    We also have to check that we have enough information to prove that ΔWVYΔVWX. See if you can spot two pairs of sides and one pair of angles that are equal between the two triangles!


    Submit your answer as:
  2. The diagram of quadrilateral WVXY is repeated for convenience:

    Now, prove that WX=VY by selecting the correct options.

    Answer:

    In :

    1. is common.
    2. WY=VX
    3. WVX=YWV=90° (given)


    WX=VY

    HINT: <no title>
    [−0 points ⇒ 6 / 6 points left]

    Prove congruency by matching up pairs of sides and angles. Use the given information and geometry facts.


    STEP: Complete the proof
    [−6 points ⇒ 0 / 6 points left]
    TIP:
    • Look for any sides or angles that the two triangles have in common.
    • Then, write down any pairs of equal sides or angles that you have been given.
    • Finally, decide what else you need in order to prove that the two triangles are congruent, according to one of the four cases of congruency.

    The correct proof is:

    In ΔWVY and ΔVWX :

    1. WV is common.
    2. WY=VX (given)
    3. WVX=YWV=90° (given)

    ΔWVYΔVWX (SAS)

    WX=VY (ΔWVYΔVWX)

    NOTE: You can extend this proof to prove that diagonals are equal in all rectangles. This is one of the properties that you should know already. Now you know why it is true!

    Submit your answer as: andandandandand

Calculate angles between parallel lines

In the diagram below, GH JK. LM is a transversal.

NOTE: A transversal is a line that cuts across two other lines.

What is the value of x?

Answer: x= °
numeric
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

STEP: <no title>
[−2 points ⇒ 0 / 2 points left]

Co-interior angles are both inside the parallel lines, on the same side of the transversal. If we highlight the angles we will see a "U" shape. This helps us to identify co-interior angles.

Since we know that co-interior angles add up to 180°, we can write:

x+118°=180°(co-int s,GHJK)x=180°118°x=62°

Therefore x=62° (co-int s, GHJK).

NOTE: Although you didn't need to give a reason here, reasons are important in geometry. You should learn these reasons and always write them in your work.

Submit your answer as:

Given diagonal properties, identify the quadrilateral

Consider the following information about the diagonals of a quadrilateral PQRS :

The diagonals of PQRS bisect the corner angles.

Select the type of quadrilateral for which this statement is definitely true.

Answer: PQRS is a:
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Think about the diagonal properties of the different quadrilaterals. Which one has the exact properties described in the question?


STEP: Recall diagonal properties
[−2 points ⇒ 0 / 2 points left]

The diagonals of all rhombuses bisect the corner angles.

NOTE: Bisect means to cut into two equal parts.

NOTE: This is true for all rhombuses, so it must also be true for squares (which are types of rhombuses).

Submit your answer as:

Recognise pairs of angles between parallel lines

In the diagram below, XYTU.  VW is a transversal.

NOTE: A transversal is a line that cuts across two other lines.

What type of angle pair is shown by the coloured angles?

Answer: The angles are .
HINT: <no title>
[−0 points ⇒ 1 / 1 points left]

Can you identify the relationship between the coloured angles?


STEP: Identify the type of angle pair
[−1 point ⇒ 0 / 1 points left]

This intersection creates alternate angles.

Alternate angles are both inside the parallel lines , but on opposite sides of the transversal. If we highlight the angles we will see a "Z" shape. This helps us identify alternate angles.


Submit your answer as:

Congruency in kites

  1. Consider the following diagram:

    Prove that ΔABCΔADC by completing the proof below.

    Answer:

    In ΔABC and ΔADC:

    1. AB=
    2. =DC
    3. is common

    ΔABCΔADC

    HINT: <no title>
    [−0 points ⇒ 6 / 6 points left]

    Use the properties of a kite to work out which sides are equal to each other.


    STEP: Use the properties of a kite to match up equal sides
    [−6 points ⇒ 0 / 6 points left]

    First, we focus on the two triangles that we need to prove congruent, and ignore the rest of the diagram.

    A kite is a quadrilateral that has two pairs of adjacent sides equal. Using this, and the information we were given in the diagram, we will highlight the sides which we know must be equal in our two triangles.

    In ΔABC and ΔADC:

    1. AB=AD (adjacent sides of kite)
    2. BC=DC (adjacent sides of kite)
    3. AC is common

    We have proved that three pairs of sides are equal, so we can use the congruency case SSS.

    ΔABCΔADC (SSS)


    Submit your answer as: andandandandand
  2. Consider the following diagram:

    You have just proved that ΔABCΔADC. Hence, determine the size of BA^C.

    Answer: BA^C= °
    numeric
    HINT: <no title>
    [−0 points ⇒ 2 / 2 points left]

    Once you have proved congruency, you know that matching angles will be equal. Which angle matches BA^C?


    STEP: Identify matching angles to determine size
    [−2 points ⇒ 0 / 2 points left]

    We have proved that ΔABCΔADC.

    Therefore, the matching angles in these two triangles must be equal.

    In particular, BA^C=DA^C.

    They are only equal because the triangles are congruent, so we use the congruency statement as our reason.

    BA^C= 46° (ΔABCΔADC)


    Submit your answer as: and

Exercises

Apply diagonal properties of a rhombus

In the diagram below, ABCD is a rhombus. Also, AD=17 and DB=30.

Determine, with reasons, the length of AE.

INSTRUCTION: Answer this question by completing the steps below.
Answer: DE= units
Also, AE^D= °
Hence, AE= units
numeric
numeric
numeric
HINT: <no title>
[−0 points ⇒ 6 / 6 points left]

What do you need to know about a triangle, before you can use the theorem of Pythagoras? Use the properties that you know about the diagonals of a rhombus.


STEP: Decide on a strategy
[−1 point ⇒ 5 / 6 points left]

We are going to use the theorem of Pythagoras in ΔAED to determine DE. To do this, we will use rhombus diagonal properties to determine the necessary sides, and to prove that ΔAED is right-angled.


STEP: Apply diagonal properties to determine DE
[−1 point ⇒ 4 / 6 points left]
The diagonals of a rhombus bisect each other.

DB=30 (given)
And, DE=EB (diags of rhombus)
DE= 15 units


STEP: Show that ΔAED is right-angled
[−2 points ⇒ 2 / 6 points left]
The diagonals of a rhombus intersect at 90°.

AE^D=90° (diags of rhombus)


STEP: Use the theorem of Pythagoras to determine the length of AE
[−2 points ⇒ 0 / 6 points left]

We have shown that ΔAED is right-angled. Now, we can use the theorem of Pythagoras to work out AE.

AE2+DE2=AD2 (Pythagoras)

AE2+152=172AE2+225=289AE2=64AE=64AE=8 units

Submit your answer as: andandandandand

Apply diagonal properties of a rhombus

In the diagram below, PQRS is a rhombus. Also, PS=15 and PR=18.

Determine, with reasons, the length of SM.

INSTRUCTION: Answer this question by completing the steps below.
Answer: PM= units
Also, PM^S= °
Hence, SM= units
numeric
numeric
numeric
HINT: <no title>
[−0 points ⇒ 6 / 6 points left]

What do you need to know about a triangle, before you can use the theorem of Pythagoras? Use the properties that you know about the diagonals of a rhombus.


STEP: Decide on a strategy
[−1 point ⇒ 5 / 6 points left]

We are going to use the theorem of Pythagoras in ΔPMS to determine PM. To do this, we will use rhombus diagonal properties to determine the necessary sides, and to prove that ΔPMS is right-angled.


STEP: Apply diagonal properties to determine PM
[−1 point ⇒ 4 / 6 points left]
The diagonals of a rhombus bisect each other.

PR=18 (given)
And, PM=MR (diags of rhombus)
PM= 9 units


STEP: Show that ΔPMS is right-angled
[−2 points ⇒ 2 / 6 points left]
The diagonals of a rhombus intersect at 90°.

PM^S=90° (diags of rhombus)


STEP: Use the theorem of Pythagoras to determine the length of SM
[−2 points ⇒ 0 / 6 points left]

We have shown that ΔPMS is right-angled. Now, we can use the theorem of Pythagoras to work out SM.

SM2+PM2=PS2 (Pythagoras)

SM2+92=152SM2+81=225SM2=144SM=144SM=12 units

Submit your answer as: andandandandand

Apply diagonal properties of a rhombus

In the diagram below, PQRS is a rhombus. Also, PS=34 and PR=32.

Determine, with reasons, the length of SM.

INSTRUCTION: Answer this question by completing the steps below.
Answer: PM= units
Also, PM^S= °
Hence, SM= units
numeric
numeric
numeric
HINT: <no title>
[−0 points ⇒ 6 / 6 points left]

What do you need to know about a triangle, before you can use the theorem of Pythagoras? Use the properties that you know about the diagonals of a rhombus.


STEP: Decide on a strategy
[−1 point ⇒ 5 / 6 points left]

We are going to use the theorem of Pythagoras in ΔPMS to determine PM. To do this, we will use rhombus diagonal properties to determine the necessary sides, and to prove that ΔPMS is right-angled.


STEP: Apply diagonal properties to determine PM
[−1 point ⇒ 4 / 6 points left]
The diagonals of a rhombus bisect each other.

PR=32 (given)
And, PM=MR (diags of rhombus)
PM= 16 units


STEP: Show that ΔPMS is right-angled
[−2 points ⇒ 2 / 6 points left]
The diagonals of a rhombus intersect at 90°.

PM^S=90° (diags of rhombus)


STEP: Use the theorem of Pythagoras to determine the length of SM
[−2 points ⇒ 0 / 6 points left]

We have shown that ΔPMS is right-angled. Now, we can use the theorem of Pythagoras to work out SM.

SM2+PM2=PS2 (Pythagoras)

SM2+162=342SM2+256=1156SM2=900SM=900SM=30 units

Submit your answer as: andandandandand

Reasoning about properties of quadrilaterals

Consider the following statement and decide if it is always true, sometimes true, or never true.

The interior angles of a quadrilateral add up to 300°.
Answer:

The statement is true.

HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Are there any quadrilaterals that make the statement true? Are there any quadrilaterals that make the statement false?


STEP: Think about whether any quadrilaterals make the statement true
[−2 points ⇒ 0 / 2 points left]

The interior angles of a quadrilateral always add up to 360°. This is because any quadrilateral can always be divided into two triangles. The angles in a triangle add up to 180°, so the angles in two triangles add up as follows: 180° + 180° = 360°.

So, anything that says something different can never be true.


Submit your answer as:

Reasoning about properties of quadrilaterals

Consider the following statement and decide if it is always true, sometimes true, or never true.

Quadrilaterals have two pairs of opposite sides that are parallel.
Answer:

The statement is true.

HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Are there any quadrilaterals that make the statement true? Are there any quadrilaterals that make the statement false?


STEP: Think about whether any quadrilaterals make the statement true
[−2 points ⇒ 0 / 2 points left]

It is clear that not all quadrilaterals have two pairs of opposite sides that are parallel. In fact, many do not, and the diagram below is an example:

NOTE: This example is called a counter-example, because it is an example that shows that the statement is not true for all cases.

But, there are some quadrilaterals that do have two pairs of opposite sides that are parallel. For example, a parallelogram is a type of quadrilateral that has two pairs of opposite sides that are parallel.

So, the statement is sometimes true.


Submit your answer as:

Reasoning about properties of quadrilaterals

Consider the following statement and decide if it is always true, sometimes true, or never true.

Kites have corner angles that are all equal to 90°.
Answer:

The statement is true.

HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Are there any quadrilaterals that make the statement true? Are there any quadrilaterals that make the statement false?


STEP: Think about whether any quadrilaterals make the statement true
[−2 points ⇒ 0 / 2 points left]

It is clear that not all kites have corner angles that are all equal to 90°. In fact, many do not, and the diagram below is an example:

NOTE: This example is called a counter-example, because it is an example that shows that the statement is not true for all cases.

But, there are some kites that do have corner angles that are all equal to 90°. For example, a square is a special type of kite that has corner angles equal to 90°.

So, the statement is sometimes true.


Submit your answer as:

Identify all diagonal properties

Think about the diagonals of a kite.

Select whether each of the following statements is true or false for all kites.

Answer:
  1. The diagonals bisect each other.
  2. Both diagonals are the same length.
  3. The diagonals intersect at 90°.
  4. The diagonals bisect the corner angles.
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

Draw a standard kite and compare the diagram to each diagonal property.


STEP: Identify the diagonal properties
[−4 points ⇒ 0 / 4 points left]

When we draw in the diagonals, we create several pairs of congruent triangles, including ΔPQTΔPST. (This is a bit harder to prove - but you should still try!) This means that T1 and T2 are equal. We also know that T1 and T2 add up to 180°. So, they must both be 90°. So, the diagonals intersect at 90°.

NOTE: One diagonal of a kite, QS, is bisected by the other. But PR is not bisected. So, we do not say that the diagonals bisect each other in a kite.

This is the only property that we can prove for certain about the diagonals of all kites. The following table shows the correct answers:

Diagonal property Definitely true for any kite?
The diagonals bisect each other. False
Both diagonals are the same length. False
The diagonals intersect at 90°. True
The diagonals bisect the corner angles. False

Submit your answer as: andandand

Identify all diagonal properties

Think about the diagonals of a kite.

Select whether each of the following statements is true or false for all kites.

Answer:
  1. The diagonals bisect each other.
  2. Both diagonals are the same length.
  3. The diagonals intersect at 90°.
  4. The diagonals bisect the corner angles.
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

Draw a standard kite and compare the diagram to each diagonal property.


STEP: Identify the diagonal properties
[−4 points ⇒ 0 / 4 points left]

When we draw in the diagonals, we create several pairs of congruent triangles, including ΔABEΔADE. (This is a bit harder to prove - but you should still try!) This means that E1 and E2 are equal. We also know that E1 and E2 add up to 180°. So, they must both be 90°. So, the diagonals intersect at 90°.

NOTE: One diagonal of a kite, BD, is bisected by the other. But AC is not bisected. So, we do not say that the diagonals bisect each other in a kite.

This is the only property that we can prove for certain about the diagonals of all kites. The following table shows the correct answers:

Diagonal property Definitely true for any kite?
The diagonals bisect each other. False
Both diagonals are the same length. False
The diagonals intersect at 90°. True
The diagonals bisect the corner angles. False

Submit your answer as: andandand

Identify all diagonal properties

Think about the diagonals of a rectangle.

Select whether each of the following statements is true or false for all rectangles.

Answer:
  1. The diagonals bisect each other.
  2. Both diagonals are the same length.
  3. The diagonals intersect at 90°.
  4. The diagonals bisect the corner angles.
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

Draw a standard rectangle and compare the diagram to each diagonal property.


STEP: Identify the diagonal properties
[−4 points ⇒ 0 / 4 points left]

The diagonals of a parallelogram bisect each other. A rectangle is a special type of parallelogram, so the diagonals of a rectangle must also bisect each other.

NOTE: Bisect means to cut into two equal parts.

When we draw in the diagonals of a rectangle, we also get ΔPRSΔQSR. (Think about how you would prove this!) This means that PR=QS . In other words, the two diagonals are the same length.

These are the only properties that we can prove for certain about the diagonals of all rectangles. The following table shows the correct answers:

Diagonal property Definitely true for any rectangle?
The diagonals bisect each other. True
Both diagonals are the same length. True
The diagonals intersect at 90°. False
The diagonals bisect the corner angles. False

Submit your answer as: andandand

Congruent triangles

Have a look at the following triangles, which are drawn to scale:

  1. Answer the two questions below.

    Answer:

    Are these triangles congruent?
    If yes, select the reason. If no, select 'not congruent.'

    HINT: <no title>
    [−0 points ⇒ 2 / 2 points left]

    The word "congruent" means "same shape". Look carefully at the triangles and the given information. Can you prove that these triangles are congruent? If so, remember to provide a correct reason.


    STEP: Examine the picture to see if the triangles are identical
    [−2 points ⇒ 0 / 2 points left]

    The word "congruent" means "identical." If the triangles have exactly the same shape, then they are congruent.

    There are four different ways to determine if two triangles are congruent.

    1. Right angle, hypotenuse, side (RHS): the two triangles are right-angled triangles, the hypotenuse of one triangle is the same length as the hypotenuse of the other triangle and one of the other pairs of corresponding sides are equal in length.
    2. Side, side, side (SSS): all three pairs of corresponding sides in the two triangles have equal lengths.
    3. Side, angle, side (SAS): two pairs of corresponding sides and the included angle are equal.
    4. Angle, angle, side (AAS): two pairs of corresponding angles are equal and one pair of corresponding sides have equal lengths.

    In this case, none of the above reasons apply: you can see that the triangles have different shapes. The triangle on the right is taller and thinner than the triangle on the left. The labels for both triangles also indicate that the sides are not the same length.

    Therefore, these two triangles are not congruent.


    Submit your answer as: and
  2. Which of the following options correctly completes the statement below?

    Answer: ΔVWUΔ
    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]
    When shapes are congruent, you must label them in the order of their corresponding angles. "Corresponding" means "matching." Look at the triangles: does U^ correspond to angle X^,Y^, or Z^?
    STEP: Compare the triangles to complete the statement
    [−1 point ⇒ 0 / 1 points left]

    When we name a pair of congruent triangles we need to pay careful attention to the order of the angles and which sides are equal: the triangles must be named according to their corresponding angles and sides. "Corresponding" means "matching."

    However, these two triangles are not congruent so there is no way to complete the statement in the question.

    The correct choice from the list is "No answer" because the shapes are not congruent.


    Submit your answer as:

Congruent triangles

Have a look at the following triangles, which are drawn to scale:

  1. Answer the two questions below.

    Answer:

    Are these triangles congruent?
    If yes, select the reason. If no, select 'not congruent.'

    HINT: <no title>
    [−0 points ⇒ 2 / 2 points left]

    The word "congruent" means "same shape". Look carefully at the triangles and the given information. Can you prove that these triangles are congruent? If so, remember to provide a correct reason.


    STEP: Examine the picture to see if the triangles are identical
    [−2 points ⇒ 0 / 2 points left]

    The word "congruent" means "identical." If the triangles have exactly the same shape, then they are congruent.

    There are four different ways to determine if two triangles are congruent.

    1. Right angle, hypotenuse, side (RHS): the two triangles are right-angled triangles, the hypotenuse of one triangle is the same length as the hypotenuse of the other triangle and one of the other pairs of corresponding sides are equal in length.
    2. Side, side, side (SSS): all three pairs of corresponding sides in the two triangles have equal lengths.
    3. Side, angle, side (SAS): two pairs of corresponding sides and the included angle are equal.
    4. Angle, angle, side (AAS): two pairs of corresponding angles are equal and one pair of corresponding sides have equal lengths.

    The sides of both triangles are labelled with x, y and z. This means that there are three pairs of corresponding and equal sides.

    Therefore, these two triangles are congruent, and the reason is SSS.


    Submit your answer as: and
  2. Which of the following options correctly completes the statement below?

    Answer: ΔNMPΔ
    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]
    When shapes are congruent, you must label them in the order of their corresponding angles. "Corresponding" means "matching." Look at the triangles: does M^ correspond to angle Q^,R^, or S^?
    STEP: Compare the triangles to complete the statement
    [−1 point ⇒ 0 / 1 points left]

    When we name a pair of congruent triangles we need to pay careful attention to the order of the angles and which sides are equal: the triangles must be named according to their corresponding angles and sides. "Corresponding" means "matching."

    The first triangle can be named in any order - we are given ΔNMP. The second triangle needs to be named so that we can read off the corresponding angles and sides: we need to ensure that the corresponding sides and angles match up.

    For example, vertex N corresponds to vertex R. Therefore, the answer must start with R. Now think about which vertex corresponds to vertex M to figure out which letter should be second in the answer.

    The correct choice to complete the statement is ΔRQS.


    Submit your answer as:

Congruent triangles

Have a look at the following triangles, which are drawn to scale:

  1. Answer the two questions below.

    Answer:

    Are these triangles congruent?
    If yes, select the reason. If no, select 'not congruent.'

    HINT: <no title>
    [−0 points ⇒ 2 / 2 points left]

    The word "congruent" means "same shape". Look carefully at the triangles and the given information. Can you prove that these triangles are congruent? If so, remember to provide a correct reason.


    STEP: Examine the picture to see if the triangles are identical
    [−2 points ⇒ 0 / 2 points left]

    The word "congruent" means "identical." If the triangles have exactly the same shape, then they are congruent.

    There are four different ways to determine if two triangles are congruent.

    1. Right angle, hypotenuse, side (RHS): the two triangles are right-angled triangles, the hypotenuse of one triangle is the same length as the hypotenuse of the other triangle and one of the other pairs of corresponding sides are equal in length.
    2. Side, side, side (SSS): all three pairs of corresponding sides in the two triangles have equal lengths.
    3. Side, angle, side (SAS): two pairs of corresponding sides and the included angle are equal.
    4. Angle, angle, side (AAS): two pairs of corresponding angles are equal and one pair of corresponding sides have equal lengths.

    The sides of both triangles are labelled with x, y and z. This means that there are three pairs of corresponding and equal sides.

    Therefore, these two triangles are congruent, and the reason is SSS.


    Submit your answer as: and
  2. Which of the following options correctly completes the statement below?

    Answer: ΔGHFΔ
    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]
    When shapes are congruent, you must label them in the order of their corresponding angles. "Corresponding" means "matching." Look at the triangles: does F^ correspond to angle J^,K^, or L^?
    STEP: Compare the triangles to complete the statement
    [−1 point ⇒ 0 / 1 points left]

    When we name a pair of congruent triangles we need to pay careful attention to the order of the angles and which sides are equal: the triangles must be named according to their corresponding angles and sides. "Corresponding" means "matching."

    The first triangle can be named in any order - we are given ΔGHF. The second triangle needs to be named so that we can read off the corresponding angles and sides: we need to ensure that the corresponding sides and angles match up.

    For example, vertex G corresponds to vertex K. Therefore, the answer must start with K. Now think about which vertex corresponds to vertex H to figure out which letter should be second in the answer.

    The correct choice to complete the statement is ΔKLJ.


    Submit your answer as:

Calculate angles between parallel lines

The diagram represents transversal line GH that intersects with the straight lines CD and EF respectively. CD EF.

The intersecting lines create different angles which are labeled for you. Carefully study the diagram and answer the questions that follow.

Answer:
  1. What is the value of x? °
  2. What is the value of y? °
  3. What is the value of z? °
numeric
numeric
numeric
HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

STEP: Determine the unknown angles
[−3 points ⇒ 0 / 3 points left]

If the two lines are parallel, the four angles around the first intersection are the same as the four angles around the second intersection. We can use this, and the fact that angles on a straight line add up to 180°, to determine the unknown angles.

Therefore:

  1. x=79°
  2. y=101°
  3. z=79°

Submit your answer as: andand

Calculate angles between parallel lines

The diagram represents transversal line LM that intersects with the straight lines GH and JK respectively. GH JK.

The intersecting lines create different angles which are labeled for you. Carefully study the diagram and answer the questions that follow.

Answer:
  1. What is the value of x? °
  2. What is the value of y? °
  3. What is the value of z? °
numeric
numeric
numeric
HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

STEP: Determine the unknown angles
[−3 points ⇒ 0 / 3 points left]

If the two lines are parallel, the four angles around the first intersection are the same as the four angles around the second intersection. We can use this, and the fact that angles on a straight line add up to 180°, to determine the unknown angles.

Therefore:

  1. x=112°
  2. y=68°
  3. z=68°

Submit your answer as: andand

Calculate angles between parallel lines

The diagram represents transversal line EF that intersects with the straight lines AB and CD respectively. AB CD.

The intersecting lines create different angles which are labeled for you. Carefully study the diagram and answer the questions that follow.

Answer:
  1. What is the value of x? °
  2. What is the value of y? °
  3. What is the value of z? °
numeric
numeric
numeric
HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

STEP: Determine the unknown angles
[−3 points ⇒ 0 / 3 points left]

If the two lines are parallel, the four angles around the first intersection are the same as the four angles around the second intersection. We can use this, and the fact that angles on a straight line add up to 180°, to determine the unknown angles.

Therefore:

  1. x=59°
  2. y=121°
  3. z=59°

Submit your answer as: andand

Proof: is the quadrilateral a parallelogram?

The figure below shows quadrilateral MNOP with MPNO and MP=NO. Diagonal NP is shown with a dashed line. M=y and O=68° ; MPN=x and NPO=61°.

  1. The steps and reasons below prove that MNOP is a parallelogram. But some steps and reasons are incomplete. Choose the correct steps and reasons from the drop down boxes to complete the proof.

    Answer:
    STEPS REASONS
    MPN=PNO Alternate interior angles are equal if MPNO
    NP=PN Common side (in triangles MNP & OPN)
    MP=NO Given
    MNP=OPN Corresponding angles in congruent triangles
    MNPO
    Definition of a parallelogram (opposite sides parallel)
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    We already know that side MP is parallel to side NO. The proof above shows that the other two sides are also parallel, based on the information given about the angles. Work your way down the proof, one step at a time, to choose the correct steps and reasons.


    STEP: Work your way down the proof, choosing the answers along the way
    [−4 points ⇒ 0 / 4 points left]

    The overall strategy of the proof in this question is to show that the sides MN and PO are parallel. If we can show that, we can conclude that the quadrilateral is a parallelogram because we already know that MP is parallel to NO. But it takes six steps to first prove that MN and PO are parallel.

    The proof starts by using the parallel sides and the diagonal. The parallel lines and the diagonal invite us to use transversal geometry: alternating interior angles, corresponding angles, cointerior angles, and so on. So we can show that two of the angles are equal. Then the proof proceeds to show (in step 4) that the triangles MNP and OPN are congruent (which means that the triangles are identical).

    The complete proof is as follows:

    Steps Reasons
    MPN=PNO Alternate interior angles are equal if MPNO
    NP=PN Common side (in triangles MNP & OPN)
    MP=NO Given
    ΔMNP=ΔOPN Side-Angle-Side for congruent triangles (SAS)
    MNP=OPN Corresponding angles in congruent triangles
    MNPO MNP & OPN are equal alternate interior angles
    MNOP is a parallelogram Definition of a parallelogram (opposite sides parallel)

    Submit your answer as: andandand
  2. Determine the value of y. The diagram is repeated here, but now with both sides labelled as parallel (which we know from Question 1).

    Answer: y= °
    numeric
    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]

    Use the fact that opposite angles in a parallelogram are equal.


    STEP: Use the fact that opposite angles in a parallelogram are equal
    [−1 point ⇒ 0 / 1 points left]

    MNOP is a parallelogram. Opposite angles of a parallelogram are equal. So M is equal to O.

    Therefore, y=68°.


    Submit your answer as:
  3. Determine the value of x.

    Answer: x= °
    numeric
    HINT: <no title>
    [−0 points ⇒ 2 / 2 points left]

    Start by writing an equation for the angles in triangle MNP. The sum of those angles, which include x, must be 180°.


    STEP: Set up an equation and solve for x
    [−2 points ⇒ 0 / 2 points left]

    We can solve this problem using the fact that angle x is inside of a triangle MNP.

    The sum of angles in a triangle is 180°, so based on the figure we can write:

    M+MPN+MNP=180°

    From Question 2 we know that y=68° (this is labelled in the diagram above). And because of alternate interior angles we also know that MNP=61°. Putting all of this together:

    M+MPN+MNP=180°68°+x+61°=180°x=180°68°61°x=51°
    NOTE: You can also solve this question using the fact that the angles in a quadrliateral have a sum of 360°. Using the fact that opposite angles in a parallelogram are equal, the equation would be:
    68°+68°+2(61°+x)=360°
    If you solve this you will also get the answer x=51°.

    The measure of angle x is 51°.


    Submit your answer as:

Proof: is the quadrilateral a parallelogram?

The figure below shows quadrilateral MNOP with MPNO and MP=NO. Diagonal NP is shown with a dashed line. M=y and O=75° ; MPN=70° and NPO=x.

  1. The steps and reasons below prove that MNOP is a parallelogram. But some steps and reasons are incomplete. Choose the correct steps and reasons from the drop down boxes to complete the proof.

    Answer:
    STEPS REASONS
    MPN=PNO Alternate interior angles are equal if MPNO
    NP=PN Common side (in triangles MNP & OPN)
    MP=NO
    ΔMNP=ΔOPN
    MNP=OPN Corresponding angles in congruent triangles
    MNP & OPN are equal alternate interior angles
    Definition of a parallelogram (opposite sides parallel)
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    We already know that side MP is parallel to side NO. The proof above shows that the other two sides are also parallel, based on the information given about the angles. Work your way down the proof, one step at a time, to choose the correct steps and reasons.


    STEP: Work your way down the proof, choosing the answers along the way
    [−4 points ⇒ 0 / 4 points left]

    The overall strategy of the proof in this question is to show that the sides MN and PO are parallel. If we can show that, we can conclude that the quadrilateral is a parallelogram because we already know that MP is parallel to NO. But it takes six steps to first prove that MN and PO are parallel.

    The proof starts by using the parallel sides and the diagonal. The parallel lines and the diagonal invite us to use transversal geometry: alternating interior angles, corresponding angles, cointerior angles, and so on. So we can show that two of the angles are equal. Then the proof proceeds to show (in step 4) that the triangles MNP and OPN are congruent (which means that the triangles are identical).

    The complete proof is as follows:

    Steps Reasons
    MPN=PNO Alternate interior angles are equal if MPNO
    NP=PN Common side (in triangles MNP & OPN)
    MP=NO Given
    ΔMNP=ΔOPN Side-Angle-Side for congruent triangles (SAS)
    MNP=OPN Corresponding angles in congruent triangles
    MNPO MNP & OPN are equal alternate interior angles
    MNOP is a parallelogram Definition of a parallelogram (opposite sides parallel)

    Submit your answer as: andandand
  2. Determine the value of y. The diagram is repeated here, but now with both sides labelled as parallel (which we know from Question 1).

    Answer: y= °
    numeric
    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]

    Use the fact that opposite angles in a parallelogram are equal.


    STEP: Use the fact that opposite angles in a parallelogram are equal
    [−1 point ⇒ 0 / 1 points left]

    MNOP is a parallelogram. Opposite angles of a parallelogram are equal. So M is equal to O.

    Therefore, y=75°.


    Submit your answer as:
  3. Determine the value of x.

    Answer: x= °
    numeric
    HINT: <no title>
    [−0 points ⇒ 2 / 2 points left]

    Start by writing an equation for the angles in triangle ONP. The sum of those angles, which include x, must be 180°.


    STEP: Set up an equation and solve for x
    [−2 points ⇒ 0 / 2 points left]

    We can solve this problem using the fact that angle x is inside of a triangle ONP.

    The sum of angles in a triangle is 180°, so based on the figure we can write:

    O+OPN+ONP=180°

    From Question 2 we know that y=75° (this is labelled in the diagram above). And because of alternate interior angles we also know that ONP=70°. Putting all of this together:

    O+OPN+ONP=180°75°+x+70°=180°x=180°75°70°x=35°
    NOTE: You can also solve this question using the fact that the angles in a quadrliateral have a sum of 360°. Using the fact that opposite angles in a parallelogram are equal, the equation would be:
    75°+75°+2(70°+x)=360°
    If you solve this you will also get the answer x=35°.

    The measure of angle x is 35°.


    Submit your answer as:

Proof: is the quadrilateral a parallelogram?

The figure below shows quadrilateral QRST with QTRS and QT=RS. Diagonal RT is shown with a dashed line. Q=63° and S=y ; QTR=48° and RTS=x.

  1. The steps and reasons below prove that QRST is a parallelogram. But some steps and reasons are incomplete. Choose the correct steps and reasons from the drop down boxes to complete the proof.

    Answer:
    STEPS REASONS
    QTR=TRS
    RT=TR Common side (in triangles QRT & STR)
    Given
    Side-Angle-Side for congruent triangles (SAS)
    QRT=STR
    QRTS QRT & STR are equal alternate interior angles
    QRST is a parallelogram Definition of a parallelogram (opposite sides parallel)
    HINT: <no title>
    [−0 points ⇒ 4 / 4 points left]

    We already know that side QT is parallel to side RS. The proof above shows that the other two sides are also parallel, based on the information given about the angles. Work your way down the proof, one step at a time, to choose the correct steps and reasons.


    STEP: Work your way down the proof, choosing the answers along the way
    [−4 points ⇒ 0 / 4 points left]

    The overall strategy of the proof in this question is to show that the sides QR and TS are parallel. If we can show that, we can conclude that the quadrilateral is a parallelogram because we already know that QT is parallel to RS. But it takes six steps to first prove that QR and TS are parallel.

    The proof starts by using the parallel sides and the diagonal. The parallel lines and the diagonal invite us to use transversal geometry: alternating interior angles, corresponding angles, cointerior angles, and so on. So we can show that two of the angles are equal. Then the proof proceeds to show (in step 4) that the triangles QRT and STR are congruent (which means that the triangles are identical).

    The complete proof is as follows:

    Steps Reasons
    QTR=TRS Alternate interior angles are equal if QTRS
    RT=TR Common side (in triangles QRT & STR)
    QT=RS Given
    ΔQRT=ΔSTR Side-Angle-Side for congruent triangles (SAS)
    QRT=STR Corresponding angles in congruent triangles
    QRTS QRT & STR are equal alternate interior angles
    QRST is a parallelogram Definition of a parallelogram (opposite sides parallel)

    Submit your answer as: andandand
  2. Determine the value of y. The diagram is repeated here, but now with both sides labelled as parallel (which we know from Question 1).

    Answer: y= °
    numeric
    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]

    Use the fact that opposite angles in a parallelogram are equal.


    STEP: Use the fact that opposite angles in a parallelogram are equal
    [−1 point ⇒ 0 / 1 points left]

    QRST is a parallelogram. Opposite angles of a parallelogram are equal. So Q is equal to S.

    Therefore, y=63°.


    Submit your answer as:
  3. Determine the value of x.

    Answer: x= °
    numeric
    HINT: <no title>
    [−0 points ⇒ 2 / 2 points left]

    Start by writing an equation for the angles in triangle SRT. The sum of those angles, which include x, must be 180°.


    STEP: Set up an equation and solve for x
    [−2 points ⇒ 0 / 2 points left]

    We can solve this problem using the fact that angle x is inside of a triangle SRT.

    The sum of angles in a triangle is 180°, so based on the figure we can write:

    S+STR+SRT=180°

    From Question 2 we know that y=63° (this is labelled in the diagram above). And because of alternate interior angles we also know that SRT=48°. Putting all of this together:

    S+STR+SRT=180°63°+x+48°=180°x=180°63°48°x=69°
    NOTE: You can also solve this question using the fact that the angles in a quadrliateral have a sum of 360°. Using the fact that opposite angles in a parallelogram are equal, the equation would be:
    63°+63°+2(48°+x)=360°
    If you solve this you will also get the answer x=69°.

    The measure of angle x is 69°.


    Submit your answer as:

Choose one correct diagonal property

ABCD below is a rhombus.

Answer:

Which of the following statements is definitely true about the diagonals of ABCD?

HINT: <no title>
[−0 points ⇒ 2 / 2 points left]
  • Diagonals go from one corner of the rhombus to the opposite corner.
  • Bisect means to cut into two equal parts.
  • Perpendicular means that the lines meet at 90°.

STEP: Identify the diagonal properties
[−2 points ⇒ 0 / 2 points left]

The diagonals of a kite intersect at 90°. A rhombus is a special type of kite, so the diagonals of a rhombus must also intersect at 90°.

The diagonals of a parallelogram bisect each other. A rhombus is a special type of parallelogram, so the diagonals of a rhombus must also bisect each other.

When we draw in the diagonals, we create several pairs of congruent triangles. In particular ΔAED, ΔCED, ΔCEB and ΔAEB are all congruent to each other. (Think about how you would prove this!)

This means that:

  • AD^E=CD^E ,
  • DC^E=BC^E ,
  • CB^E=AB^E , and
  • BA^E=DA^E .

The corner angles are all cut into two equal parts by the diagonals, so the diagonals bisect the corner angles.

These are the only properties that we can prove for certain about the diagonals of all rhombuses.

Diagonal property Definitely true for any rhombus?
The diagonals bisect each other.
Both diagonals are the same length.
The diagonals intersect at 90°.
The diagonals bisect the corner angles.

Of the options given, only C is definitely true for all rhombuses.


Submit your answer as:

Choose one correct diagonal property

ABCD below is a rectangle.

Answer:

Which of the following statements is definitely true about the diagonals of ABCD?

HINT: <no title>
[−0 points ⇒ 2 / 2 points left]
  • Diagonals go from one corner of the rectangle to the opposite corner.
  • Bisect means to cut into two equal parts.
  • Perpendicular means that the lines meet at 90°.

STEP: Identify the diagonal properties
[−2 points ⇒ 0 / 2 points left]

The diagonals of a parallelogram bisect each other. A rectangle is a special type of parallelogram, so the diagonals of a rectangle must also bisect each other.

NOTE: Bisect means to cut into two equal parts.

When we draw in the diagonals of a rectangle, we also get ΔACDΔBDC. (Think about how you would prove this!) This means that AC=BD . In other words, the two diagonals are the same length.

These are the only properties that we can prove for certain about the diagonals of all rectangles.

Diagonal property Definitely true for any rectangle?
The diagonals bisect each other.
Both diagonals are the same length.
The diagonals intersect at 90°.
The diagonals bisect the corner angles.

Of the options given, only C is definitely true for all rectangles.


Submit your answer as:

Choose one correct diagonal property

ABCD below is a square.

Answer:

Which of the following statements is definitely true about the diagonals of ABCD?

HINT: <no title>
[−0 points ⇒ 2 / 2 points left]
  • Diagonals go from one corner of the square to the opposite corner.
  • Bisect means to cut into two equal parts.
  • Perpendicular means that the lines meet at 90°.

STEP: Identify the diagonal properties
[−2 points ⇒ 0 / 2 points left]

A square is a special type of rectangle and a special type of rhombus. So it will have all of the diagonal properties of rectangles and of rhombuses.

The following table sums up all of the diagonal properties of a square:

Diagonal property Definitely true for any square?
The diagonals bisect each other.
Both diagonals are the same length.
The diagonals intersect at 90°.
The diagonals bisect the corner angles.

Of the options given, only C is definitely true for all squares.


Submit your answer as:

Apply diagonal properties

In the diagram below, DC=24 units,DE=7 units, and EC=17 units.

Answer the following questions:

  1. What type of quadrilateral is ABCD? (Give the most specific name.)
  2. Determine the length of EB.
INSTRUCTION: You do not need to give any reasons for your answers in this question.
Answer:
  1. ABCD is a .
  2. EB= units
numeric
HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

Think about the diagonal properties of ABCD. How can these help you to determine EB?


STEP: Identify ABCD
[−1 point ⇒ 2 / 3 points left]

The most specific name for ABCD is a parallelogram. Even though ABCD is a type of trapezium (for example), this is not the most specific name for ABCD.

TIP: When asked to give the name of a quadrilateral, always give the most specific name.

STEP: Apply diagonal properties of a parallelogram to determine EB
[−2 points ⇒ 0 / 3 points left]

Remember the diagonal properties of a parallelogram:

Diagonal property Definitely true for a parallelogram?
The diagonals bisect each other.
Both diagonals are the same length.
The diagonals intersect at 90°.
The diagonals bisect the corner angles.

The diagonals of a parallelogram bisect each other. This means that EB = DE. Therefore, EB=7 units.



Submit your answer as: and

Apply diagonal properties

In the diagram below, PQ^T=21° and PS^T=102°.

Answer the following questions:

  1. What type of quadrilateral is PQRS? (Give the most specific name.)
  2. Determine the size of TS^R.
INSTRUCTION: You do not need to give any reasons for your answers in this question.
Answer:
  1. PQRS is a .
  2. TS^R= °
numeric
HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

Think about the diagonal properties of PQRS. How can these help you to determine TS^R?


STEP: Identify PQRS
[−1 point ⇒ 2 / 3 points left]

The most specific name for PQRS is a parallelogram. Even though PQRS is a type of trapezium (for example), this is not the most specific name for PQRS.

TIP: When asked to give the name of a quadrilateral, always give the most specific name.

STEP: Apply diagonal properties of a parallelogram to determine TS^R
[−2 points ⇒ 0 / 3 points left]

Remember the diagonal properties of a parallelogram:

Diagonal property Definitely true for a parallelogram?
The diagonals bisect each other.
Both diagonals are the same length.
The diagonals intersect at 90°.
The diagonals bisect the corner angles.

The diagonals of a parallelogram do not necessarily bisect the corner angles, so we cannot assume that TS^R is equal to PS^T. In fact, we must use the fact that alternate angles on parallel lines are equal to deduce that TS^R = PQ^T = 21°.



Submit your answer as: and

Apply diagonal properties

In the diagram below, AE=4 and AC^D=27°.

Answer the following questions:

  1. What type of quadrilateral is ABCD? (Give the most specific name.)
  2. Determine the size of AC^B.
INSTRUCTION: You do not need to give any reasons for your answers in this question.
Answer:
  1. ABCD is a .
  2. AC^B= °
numeric
HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

Think about the diagonal properties of ABCD. How can these help you to determine AC^B?


STEP: Identify ABCD
[−1 point ⇒ 2 / 3 points left]

The most specific name for ABCD is a rectangle. Even though ABCD is a type of parallelogram (for example), this is not the most specific name for ABCD.

TIP: When asked to give the name of a quadrilateral, always give the most specific name.

STEP: Apply diagonal properties of a rectangle to determine AC^B
[−2 points ⇒ 0 / 3 points left]

Remember the diagonal properties of a rectangle:

Diagonal property Definitely true for a rectangle?
The diagonals bisect each other.
Both diagonals are the same length.
The diagonals intersect at 90°.
The diagonals bisect the corner angles.

The diagonals of a rectangle do not necessarily bisect the corner angles. So, we cannot assume that AC^B = AC^D. Instead, we must use the fact that BC^D=90°:

BC^D=90° and AC^D =27° (both given)
Also,AC^B+AC^D=BC^D (from the diagram)

AC^B+27°=90°AC^B=63°

Submit your answer as: and

Congruent triangles

In the diagram below, ΔUVWΔUXW. Also, UWXV while VW=15 and UW=12.

  1. Calculate the value of x.
  2. Determine the length of XV.
Answer:
  1. x= units
  2. XV= units
numeric
numeric
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

The triangles are congruent. That means they are identical. Start by using this fact to label all the sides which are not already labelled.


STEP: Label everything you can in the figure
[−1 point ⇒ 3 / 4 points left]

The first question asks us to determine the length of the side of the diagram labelled x. The x is attached to side UX.

Based on the labels in the diagram, we do not know enough to calculate x immediately. However, the question states that the two triangles are congruent. That means they are identical. We can use that information to label all the sides which are not already labelled. The question also states that UWXV, which means that there are two right angles. We should label those right angles too.

Now we can see two right-angled triangles. And we can see which sides are equal to each other. Both of the triangles include a side labelled x.


STEP: Use the theorem of Pythagoras to find the value of x
[−2 points ⇒ 1 / 4 points left]

Now calculate x using the theorem of Pythagoras. We will focus on triangle UXW, but you can use either one, because they are congruent.

In ΔUXW:XW=15(ΔUVWΔUXW)(15)2=x2+(12)2(Pythagoras)225=x2+144225144=x2±81=x9=x

The length of UX is 9 units.


STEP: Find the length of XV
[−1 point ⇒ 0 / 4 points left]

For the second question, we need to find the length of segment XV. This segment is twice as long as x, which we calculated above. So XV=2(9)=18 units.

The correct answers are:

  1. The length of side x is 9 units.
  2. The length of XV is 18 units.

Submit your answer as: and

Congruent triangles

In the diagram below, ΔABCΔADC. Also, ACDB while AB=15 and AC=20.

  1. Calculate the value of x.
  2. Determine the length of DB.
Answer:
  1. x= units
  2. DB= units
numeric
numeric
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

The triangles are congruent. That means they are identical. Start by using this fact to label all the sides which are not already labelled.


STEP: Label everything you can in the figure
[−1 point ⇒ 3 / 4 points left]

The first question asks us to determine the length of the side of the diagram labelled x. The x is attached to side CD.

Based on the labels in the diagram, we do not know enough to calculate x immediately. However, the question states that the two triangles are congruent. That means they are identical. We can use that information to label all the sides which are not already labelled. The question also states that ACDB, which means that there are two right angles. We should label those right angles too.

Now we can see two right-angled triangles. And we can see which sides are equal to each other. Both of the triangles include a side labelled x.


STEP: Use the theorem of Pythagoras to find the value of x
[−2 points ⇒ 1 / 4 points left]

Now calculate x using the theorem of Pythagoras. We will focus on triangle ADC, but you can use either one, because they are congruent.

In ΔADC:AD=15(ΔABCΔADC)x2=(15)2+(20)2(Pythagoras)x2=225+400x=±625x=25

The length of CD is 25 units.


STEP: Find the length of DB
[−1 point ⇒ 0 / 4 points left]

For the second question, we need to find the length of segment DB. This segment is made of DA and AB. So we can add them together to get the total length: DB=15+15=30 units.

The correct answers are:

  1. The length of side x is 25 units.
  2. The length of DB is 30 units.

Submit your answer as: and

Congruent triangles

In the diagram below, ΔMNPΔMQP. Also, MPQN while MN=9 and MP=12.

  1. Calculate the value of x.
  2. Determine the length of QN.
Answer:
  1. x= units
  2. QN= units
numeric
numeric
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

The triangles are congruent. That means they are identical. Start by using this fact to label all the sides which are not already labelled.


STEP: Label everything you can in the figure
[−1 point ⇒ 3 / 4 points left]

The first question asks us to determine the length of the side of the diagram labelled x. The x is attached to side PQ.

Based on the labels in the diagram, we do not know enough to calculate x immediately. However, the question states that the two triangles are congruent. That means they are identical. We can use that information to label all the sides which are not already labelled. The question also states that MPQN, which means that there are two right angles. We should label those right angles too.

Now we can see two right-angled triangles. And we can see which sides are equal to each other. Both of the triangles include a side labelled x.


STEP: Use the theorem of Pythagoras to find the value of x
[−2 points ⇒ 1 / 4 points left]

Now calculate x using the theorem of Pythagoras. We will focus on triangle MQP, but you can use either one, because they are congruent.

In ΔMQP:MQ=9(ΔMNPΔMQP)x2=(9)2+(12)2(Pythagoras)x2=81+144x=±225x=15

The length of PQ is 15 units.


STEP: Find the length of QN
[−1 point ⇒ 0 / 4 points left]

For the second question, we need to find the length of segment QN. This segment is made of QM and MN. So we can add them together to get the total length: QN=9+9=18 units.

The correct answers are:

  1. The length of side x is 15 units.
  2. The length of QN is 18 units.

Submit your answer as: and

Definitions of quadrilaterals

  1. Select the most correct definition for a parallelogram.

    Answer: A parallelogram is a:
    HINT: <no title>
    [−0 points ⇒ 2 / 2 points left]

    You need to choose the definition that lists all the properties which make the shape what it is. None of the essential properties should be missing from your definition. You also must not include any unnecessary properties.


    STEP: Choose the definition with the exact properties
    [−2 points ⇒ 0 / 2 points left]

    We use the properties of shapes to define them. We need to choose the definition that lists all the properties which make the shape what it is. None of the essential properties should be missing from the definition. Also, the definition must not include any unnecessary properties.

    • A quadrilateral with two pairs of opposite sides parallel, adjacent sides equal, and all corners 90° is a square.
    • "Tilted rectangle" is not a useful way to think about a parallelogram. It suggests that a parallelogram is a type of rectangle, but this is not true.
    • A quadrilateral with at least one pair of opposite sides parallel is a trapezium.
    • A parallelogram is a quadrilateral with two pairs of opposite sides parallel.

    This is a diagram of a parallelogram with the definition properties drawn in:


    Submit your answer as:
  2. Complete the statement below:

    Answer:

    A has all of the same properties as a parallelogram (plus some more), so it is a special type of parallelogram.

    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]

    Look at the definition for a parallelogram, and ask yourself which quadrilateral on the list has these same properties. It can have extra properties, but it cannot have less.


    STEP: Choose the quadrilateral that has at least the same properties as a parallelogram
    [−1 point ⇒ 0 / 1 points left]

    A rectangle has two pairs of opposite sides parallel. This means that a rectangle meets the definition of a parallelogram, so we say that it is a special type of parallelogram.

    NOTE: It does not matter that a rectangle has extra properties: what matters is that it has enough properties to be called a parallelogram.

    Submit your answer as:

Definitions of quadrilaterals

  1. Select the most correct definition for a kite.

    Answer: A kite is a:
    HINT: <no title>
    [−0 points ⇒ 2 / 2 points left]

    You need to choose the definition that lists all the properties which make the shape what it is. None of the essential properties should be missing from your definition. You also must not include any unnecessary properties.


    STEP: Choose the definition with the exact properties
    [−2 points ⇒ 0 / 2 points left]

    We use the properties of shapes to define them. We need to choose the definition that lists all the properties which make the shape what it is. None of the essential properties should be missing from the definition. Also, the definition must not include any unnecessary properties.

    • A quadrilateral with at least one pair of opposite sides parallel is a trapezium.
    • A kite is a quadrilateral with two pairs of adjacent sides equal.
    • We do not refer to any quadrilaterals as "diamonds". This is an informal name. As we progress further in mathematics, we need to start to use the correct terms.
    • A quadrilateral with two pairs of opposite sides parallel and adjacent sides equal is a rhombus.

    This is a diagram of a kite with the definition properties drawn in:


    Submit your answer as:
  2. Complete the statement below:

    Answer:

    A has all of the same properties as a kite (plus some more), so it is a special type of kite.

    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]

    Look at the definition for a kite, and ask yourself which quadrilateral on the list has these same properties. It can have extra properties, but it cannot have less.


    STEP: Choose the quadrilateral that has at least the same properties as a kite
    [−1 point ⇒ 0 / 1 points left]

    A square has two pairs of adjacent sides equal. This means that a square meets the definition of a kite, so we say that it is a special type of kite.

    NOTE: It does not matter that a square has extra properties: what matters is that it has enough properties to be called a kite.

    Submit your answer as:

Definitions of quadrilaterals

  1. Select the most correct definition for a rhombus.

    Answer: A rhombus is a:
    HINT: <no title>
    [−0 points ⇒ 2 / 2 points left]

    You need to choose the definition that lists all the properties which make the shape what it is. None of the essential properties should be missing from your definition. You also must not include any unnecessary properties.


    STEP: Choose the definition with the exact properties
    [−2 points ⇒ 0 / 2 points left]

    We use the properties of shapes to define them. We need to choose the definition that lists all the properties which make the shape what it is. None of the essential properties should be missing from the definition. Also, the definition must not include any unnecessary properties.

    • A quadrilateral with two pairs of opposite sides parallel, adjacent sides equal, and all corners 90° is a square.
    • "Tilted square" is not a useful way to think about a rhombus. It suggests that a rhombus is a type of square, but this is not true.
    • A quadrilateral with two pairs of opposite sides parallel is a parallelogram.
    • A rhombus is a quadrilateral with adjacent sides equal and two pairs of opposite sides parallel.

    This is a diagram of a rhombus with the definition properties drawn in:


    Submit your answer as:
  2. Complete the statement below:

    Answer:

    A has all of the same properties as a rhombus (plus some more), so it is a special type of rhombus.

    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]

    Look at the definition for a rhombus, and ask yourself which quadrilateral on the list has these same properties. It can have extra properties, but it cannot have less.


    STEP: Choose the quadrilateral that has at least the same properties as a rhombus
    [−1 point ⇒ 0 / 1 points left]

    A square has adjacent sides equal and two pairs of opposite sides parallel. This means that a square meets the definition of a rhombus, so we say that it is a special type of rhombus.

    NOTE: It does not matter that a square has extra properties: what matters is that it has enough properties to be called a rhombus.

    Submit your answer as:

Opposites of a parallelogram

Answer the following questions about the parallelogram XWVU. The parallelogram has the following sides and angles:

X^=y, W^=108°, V^=72°, and U^=x. The sides are XW=w, WV=6, VU=5, and XU=6.

Answer:
  1. What is the size of x? °
  2. What is the size of y? °
  3. What is the length of side w? units
numeric
numeric
numeric
HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

You do not need to calculate anything. You can read all three answers from the diagram. You need to use the properties of sides and angles for parallelograms.


STEP: Find the unknown angles and the length of the missing side
[−3 points ⇒ 0 / 3 points left]

We can answer this question based on the properties of parallelograms.

There are two properties of parallelograms which are useful in this question:

  • Opposite angles in a parallelogram are equal
  • Opposite sides of parallelograms have equal lengths

Therefore:

  • x=108°
  • y=72°
  • w=5units

Submit your answer as: andand

Opposites of a parallelogram

Answer the following questions about the parallelogram ABCD. The parallelogram has the following sides and angles:

A^=68°, B^=y, C^=x, and D^=112°. The sides are AB=w, BC=5, CD=4, and AD=5.

Answer:
  1. What is the size of x? °
  2. What is the size of y? °
  3. What is the length of side w? units
numeric
numeric
numeric
HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

You do not need to calculate anything. You can read all three answers from the diagram. You need to use the properties of sides and angles for parallelograms.


STEP: Find the unknown angles and the length of the missing side
[−3 points ⇒ 0 / 3 points left]

We can answer this question based on the properties of parallelograms.

There are two properties of parallelograms which are useful in this question:

  • Opposite angles in a parallelogram are equal
  • Opposite sides of parallelograms have equal lengths

Therefore:

  • x=68°
  • y=112°
  • w=4units

Submit your answer as: andand

Opposites of a parallelogram

Answer the following questions about the parallelogram XWVU. The parallelogram has the following sides and angles:

X^=x, W^=113°, V^=67°, and U^=y. The sides are XW=4, WV=w, VU=4, and XU=8.

Answer:
  1. What is the size of x? °
  2. What is the size of y? °
  3. What is the length of side w? units
numeric
numeric
numeric
HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

You do not need to calculate anything. You can read all three answers from the diagram. You need to use the properties of sides and angles for parallelograms.


STEP: Find the unknown angles and the length of the missing side
[−3 points ⇒ 0 / 3 points left]

We can answer this question based on the properties of parallelograms.

There are two properties of parallelograms which are useful in this question:

  • Opposite angles in a parallelogram are equal
  • Opposite sides of parallelograms have equal lengths

Therefore:

  • x=67°
  • y=113°
  • w=8units

Submit your answer as: andand

Identifying quadrilaterals

Which of the following are parallelograms?

A B
C D
Answer: The parallelograms are .
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Which of these quadrilaterals have the properties which meet the definition of a parallelograms? They could have extra properties as well - that does not mean they are not parallelograms.


STEP: Identify the quadrilaterals that meet the definition of a parallelogram
[−2 points ⇒ 0 / 2 points left]
A parallelogram is a quadrilateral with two pairs of opposite sides that are parallel.

We can easily recognise Quadrilateral C as a parallelogram. However, this is not the only type of parallelogram in the options.

Quadrilaterals A and D each have two pairs of opposite sides that are parallel. So, A and D are also types of parallelograms. This is true even though the exact name for A is a rhombus, and the exact name for D is a square (because they have a few extra properties too).


Submit your answer as:

Identifying quadrilaterals

Which of the following are parallelograms?

A B
C D
Answer: The parallelograms are .
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Which of these quadrilaterals have the properties which meet the definition of a parallelograms? They could have extra properties as well - that does not mean they are not parallelograms.


STEP: Identify the quadrilaterals that meet the definition of a parallelogram
[−2 points ⇒ 0 / 2 points left]
A parallelogram is a quadrilateral with two pairs of opposite sides that are parallel.

We can easily recognise Quadrilateral D as a parallelogram. However, this is not the only type of parallelogram in the options.

Quadrilaterals A, B, and C each have two pairs of opposite sides that are parallel. So, A, B, and C are also types of parallelograms. This is true even though the exact name for A is a square, the exact name for B is a rhombus, and the exact name for C is a rectangle (because they have a few extra properties too).


Submit your answer as:

Identifying quadrilaterals

Which of the following are parallelograms?

A B
C D
Answer: The parallelograms are .
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Which of these quadrilaterals have the properties which meet the definition of a parallelograms? They could have extra properties as well - that does not mean they are not parallelograms.


STEP: Identify the quadrilaterals that meet the definition of a parallelogram
[−2 points ⇒ 0 / 2 points left]
A parallelogram is a quadrilateral with two pairs of opposite sides that are parallel.

We can easily recognise Quadrilateral B as a parallelogram. However, this is not the only type of parallelogram in the options.

Quadrilateral C has two pairs of opposite sides that are parallel. So, it is also a type of parallelogram. This is true even though its exact name is a rhombus (because it has a few extra properties too).


Submit your answer as:

Angles on a straight line

Line ST represents angles on one side of a straight line. a = 50° , b = x and c = 66°.

Answer the following questions about the diagram:

  1. What is the value of x?
  2. What type of angle is represented by x?
Answer:
  1. x= °
  2. The angle is .
numeric
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

It can be shown that the measurement of a straight angle is 180° or π radians.


STEP: Use the fact that angles on a straight line add to 180°
[−2 points ⇒ 0 / 2 points left]

Flat or straight angles are formed when the legs are pointing in exactly opposite directions. The two legs then form a single straight line through the vertex of the angle. Angles in a straight line add up to 180°.

a+b+c=180°50°+x+66°=180°x=64°

Submit your answer as: and

Angles on a straight line

Line YZ represents angles on one side of a straight line. a = 62° , b = 40° and c = x.

Answer the following questions about the diagram:

  1. What is the value of x?
  2. What type of angle is represented by x?
Answer:
  1. x= °
  2. The angle is .
numeric
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

It can be shown that the measurement of a straight angle is 180° or π radians.


STEP: Use the fact that angles on a straight line add to 180°
[−2 points ⇒ 0 / 2 points left]

Flat or straight angles are formed when the legs are pointing in exactly opposite directions. The two legs then form a single straight line through the vertex of the angle. Angles in a straight line add up to 180°.

a+b+c=180°62°+40°+x=180°x=78°

Submit your answer as: and

Angles on a straight line

Line ST represents angles on one side of a straight line. a = 69° , b = 32° and c = x.

Answer the following questions about the diagram:

  1. What is the value of x?
  2. What type of angle is represented by x?
Answer:
  1. x= °
  2. The angle is .
numeric
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

It can be shown that the measurement of a straight angle is 180° or π radians.


STEP: Use the fact that angles on a straight line add to 180°
[−2 points ⇒ 0 / 2 points left]

Flat or straight angles are formed when the legs are pointing in exactly opposite directions. The two legs then form a single straight line through the vertex of the angle. Angles in a straight line add up to 180°.

a+b+c=180°69°+32°+x=180°x=79°

Submit your answer as: and

Angles on a straight line

Examine the diagram below, and answer the questions that follow.

  1. Calculate x.
  2. What type of angle is represented with x?
Answer:
  1. x= °
  2. The angle is: .
numeric
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

The angles must have a sum of 180°. Write an equation based on this information and solve for x.


STEP: Write an equation to summarise the angles
[−1 point ⇒ 1 / 2 points left]

The three angles must have a sum of 180° because BC is a straight line. We can write an equation based on this information:

45°+x+45°=180°(s on a str line)

STEP: Solve the equation and deterimine the type of the angle
[−1 point ⇒ 0 / 2 points left]

Now we can solve the equation. Once we have the answer we can determine the type of angle.

45°+x+45°=180°x=90°

The complete diagram, with all three angles known, is:

The missing angle is 90°, which is a right angle.


Submit your answer as: and

Angles on a straight line

Examine the diagram below, and answer the questions that follow.

  1. Calculate x.
  2. What type of angle is represented with x?
Answer:
  1. x= °
  2. The angle is: .
numeric
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

The angles must have a sum of 180°. Write an equation based on this information and solve for x.


STEP: Write an equation to summarise the angles
[−1 point ⇒ 1 / 2 points left]

The three angles must have a sum of 180° because BC is a straight line. We can write an equation based on this information:

x+85°+50°=180°(s on a str line)

STEP: Solve the equation and deterimine the type of the angle
[−1 point ⇒ 0 / 2 points left]

Now we can solve the equation. Once we have the answer we can determine the type of angle.

x+85°+50°=180°x=45°

The complete diagram, with all three angles known, is:

The missing angle is 45°, which is acute.


Submit your answer as: and

Angles on a straight line

Examine the diagram below, and answer the questions that follow.

  1. Calculate x.
  2. What type of angle is represented with x?
Answer:
  1. x= °
  2. The angle is: .
numeric
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

The angles must have a sum of 180°. Write an equation based on this information and solve for x.


STEP: Write an equation to summarise the angles
[−1 point ⇒ 1 / 2 points left]

The three angles must have a sum of 180° because YZ is a straight line. We can write an equation based on this information:

x+63°+60°=180°(s on a str line)

STEP: Solve the equation and deterimine the type of the angle
[−1 point ⇒ 0 / 2 points left]

Now we can solve the equation. Once we have the answer we can determine the type of angle.

x+63°+60°=180°x=57°

The complete diagram, with all three angles known, is:

The missing angle is 57°, which is acute.


Submit your answer as: and

Prove the quadrilateral is a parallelogram

The figure below shows quadrilateral XWVU, which is a parallelogram. The 4 dashed red lines bisect each vertex (angle) of the parallelogram (angles X,W,V and U). So X1=X2, W1=W2, and so on. These dashed lines form quadrilateral EFGH inside of parallelogram XWVU.

Note the diagram is drawn to scale.

The table below shows a proof that quadrilaterial EFGH is a parallelogram. But 4 steps and 4 reasons are missing from the proof. Complete the proof by choosing the correct steps and reasons from each of the drop down boxes.

Answer:
STEPS REASONS
X=V Opposite angles in a parallelogram are equal
Angles created by bisectors of equal angles are equal
W=U
W2=U1 and W1=U2 Angles created by bisectors of equal angles are equal
XU=WV Opposite sides of a parallelogram are equal
In quadrilateral EFGH:E2=G2 Congruent triangles have equal angles (ΔXUEΔVGW)
Opposite sides of a parallelogram are equal
ΔXHWΔVFU
H1=F1 Congruent triangles have equal angles (ΔXHWΔVFU)
H1=H2 and F1=F2 Vertically opposite angles are equal
HINT: <no title>
[−0 points ⇒ 8 / 8 points left]

You should start at the top of the proof and work your way down. Try to understand each step and each reason, because you will need to know how the proof is developing from the top to the bottom.


STEP: Work down the proof and select the steps and reasons along the way
[−8 points ⇒ 0 / 8 points left]

The proof is about the quadrilateral in the middle of the figure, EFGH. The steps of the proof need to show that this quadrilateral is a parallelogram. We need to complete the proof by selecting the correct steps and reasons.

The question states that quadrilateral XWVU is a parallelogram. We know lots of facts about parallelograms. The question also states that the red dashed lines bisect the angles at each vertex. (This means the line splits the angle into two equal parts.) It is a good idea to mark everything we know on the diagram itself. The diagram below shows a number of new labels for equal angles and equal sides.

The strategy used in this proof can be summarised as follows: we want to prove that EFGH is a parallelogram. We can do that if we can show that the opposite angles inside EFGH are equal. So the proof shows that there are congruent triangles in the diagram. That leads to congruent angles. And from those congruent angles we can conclude that EFGH has two pairs of opposite angles which are equal.

NOTE: There are other ways to prove that quadrilateral EFGH is a parallelogram: this is not the only way to do it. On a test or exam, you might use a different set of steps and reasons. But for this question, you must find the steps and reasons which fit within the proof as it is presented.

The table below shows the correct answer choices (in the green blocks).

STEPS REASONS
X=V Opposite angles in a parallelogram are equal
X2=V1 and X1=V2 Angles created by bisectors of equal angles are equal
W=U Opposite angles in a parallelogram are equal
W2=U1 and W1=U2 Angles created by bisectors of equal angles are equal
XU=WV Opposite sides of a parallelogram are equal
ΔXUEΔVGW Congruent triangles (Angle-Side-Angle)
In quadrilateral EFGH:E2=G2 Congruent triangles have equal angles (ΔXUEΔVGW)
XW=UV Opposite sides of a parallelogram are equal
ΔXHWΔVFU Congruent triangles (Angle-Side-Angle)
H1=F1 Congruent triangles have equal angles (ΔXHWΔVFU)
H1=H2 and F1=F2 Vertically opposite angles are equal
EFGH is a parallelogram Opposite angles in a parallelogram are equal

Submit your answer as: andandandandandandand

Prove the quadrilateral is a parallelogram

The figure below shows quadrilateral ABCD, which is a parallelogram. The 4 dashed red lines bisect each vertex (angle) of the parallelogram (angles A,B,C and D). So A1=A2, B1=B2, and so on. These dashed lines form quadrilateral MNOP inside of parallelogram ABCD.

Note the diagram is drawn to scale.

The table below shows a proof that quadrilaterial MNOP is a parallelogram. But 4 steps and 4 reasons are missing from the proof. Complete the proof by choosing the correct steps and reasons from each of the drop down boxes.

Answer:
STEPS REASONS
Opposite angles in a parallelogram are equal
A2=C1 and A1=C2 Angles created by bisectors of equal angles are equal
B=D Opposite angles in a parallelogram are equal
Angles created by bisectors of equal angles are equal
AD=BC
ΔADMΔCOB Congruent triangles (Angle-Side-Angle)
Congruent triangles have equal angles (ΔADMΔCOB)
AB=DC
P1=N1 Congruent triangles have equal angles (ΔAPBΔCND)
P1=P2 and N1=N2
MNOP is a parallelogram Opposite angles in a parallelogram are equal
HINT: <no title>
[−0 points ⇒ 8 / 8 points left]

You should start at the top of the proof and work your way down. Try to understand each step and each reason, because you will need to know how the proof is developing from the top to the bottom.


STEP: Work down the proof and select the steps and reasons along the way
[−8 points ⇒ 0 / 8 points left]

The proof is about the quadrilateral in the middle of the figure, MNOP. The steps of the proof need to show that this quadrilateral is a parallelogram. We need to complete the proof by selecting the correct steps and reasons.

The question states that quadrilateral ABCD is a parallelogram. We know lots of facts about parallelograms. The question also states that the red dashed lines bisect the angles at each vertex. (This means the line splits the angle into two equal parts.) It is a good idea to mark everything we know on the diagram itself. The diagram below shows a number of new labels for equal angles and equal sides.

The strategy used in this proof can be summarised as follows: we want to prove that MNOP is a parallelogram. We can do that if we can show that the opposite angles inside MNOP are equal. So the proof shows that there are congruent triangles in the diagram. That leads to congruent angles. And from those congruent angles we can conclude that MNOP has two pairs of opposite angles which are equal.

NOTE: There are other ways to prove that quadrilateral MNOP is a parallelogram: this is not the only way to do it. On a test or exam, you might use a different set of steps and reasons. But for this question, you must find the steps and reasons which fit within the proof as it is presented.

The table below shows the correct answer choices (in the green blocks).

STEPS REASONS
A=C Opposite angles in a parallelogram are equal
A2=C1 and A1=C2 Angles created by bisectors of equal angles are equal
B=D Opposite angles in a parallelogram are equal
B2=D1 and B1=D2 Angles created by bisectors of equal angles are equal
AD=BC Opposite sides of a parallelogram are equal
ΔADMΔCOB Congruent triangles (Angle-Side-Angle)
In quadrilateral MNOP:M2=O2 Congruent triangles have equal angles (ΔADMΔCOB)
AB=DC Opposite sides of a parallelogram are equal
ΔAPBΔCND Congruent triangles (Angle-Side-Angle)
P1=N1 Congruent triangles have equal angles (ΔAPBΔCND)
P1=P2 and N1=N2 Vertically opposite angles are equal
MNOP is a parallelogram Opposite angles in a parallelogram are equal

Submit your answer as: andandandandandandand

Prove the quadrilateral is a parallelogram

The figure below shows quadrilateral QRST, which is a parallelogram. The 4 dashed red lines bisect each vertex (angle) of the parallelogram (angles Q,R,S and T). So Q1=Q2, R1=R2, and so on. These dashed lines form quadrilateral JKLM inside of parallelogram QRST.

Note the diagram is drawn to scale.

The table below shows a proof that quadrilaterial JKLM is a parallelogram. But 4 steps and 4 reasons are missing from the proof. Complete the proof by choosing the correct steps and reasons from each of the drop down boxes.

Answer:
STEPS REASONS
Q=S Opposite angles in a parallelogram are equal
Q2=S1 and Q1=S2 Angles created by bisectors of equal angles are equal
R=T Opposite angles in a parallelogram are equal
R2=T1 and R1=T2 Angles created by bisectors of equal angles are equal
Opposite sides of a parallelogram are equal
ΔQTJΔSLR
In quadrilateral JKLM:J2=L2 Congruent triangles have equal angles (ΔQTJΔSLR)
QR=TS Opposite sides of a parallelogram are equal
ΔQMRΔSKT
Opposite angles in a parallelogram are equal
HINT: <no title>
[−0 points ⇒ 8 / 8 points left]

You should start at the top of the proof and work your way down. Try to understand each step and each reason, because you will need to know how the proof is developing from the top to the bottom.


STEP: Work down the proof and select the steps and reasons along the way
[−8 points ⇒ 0 / 8 points left]

The proof is about the quadrilateral in the middle of the figure, JKLM. The steps of the proof need to show that this quadrilateral is a parallelogram. We need to complete the proof by selecting the correct steps and reasons.

The question states that quadrilateral QRST is a parallelogram. We know lots of facts about parallelograms. The question also states that the red dashed lines bisect the angles at each vertex. (This means the line splits the angle into two equal parts.) It is a good idea to mark everything we know on the diagram itself. The diagram below shows a number of new labels for equal angles and equal sides.

The strategy used in this proof can be summarised as follows: we want to prove that JKLM is a parallelogram. We can do that if we can show that the opposite angles inside JKLM are equal. So the proof shows that there are congruent triangles in the diagram. That leads to congruent angles. And from those congruent angles we can conclude that JKLM has two pairs of opposite angles which are equal.

NOTE: There are other ways to prove that quadrilateral JKLM is a parallelogram: this is not the only way to do it. On a test or exam, you might use a different set of steps and reasons. But for this question, you must find the steps and reasons which fit within the proof as it is presented.

The table below shows the correct answer choices (in the green blocks).

STEPS REASONS
Q=S Opposite angles in a parallelogram are equal
Q2=S1 and Q1=S2 Angles created by bisectors of equal angles are equal
R=T Opposite angles in a parallelogram are equal
R2=T1 and R1=T2 Angles created by bisectors of equal angles are equal
QT=RS Opposite sides of a parallelogram are equal
ΔQTJΔSLR Congruent triangles (Angle-Side-Angle)
In quadrilateral JKLM:J2=L2 Congruent triangles have equal angles (ΔQTJΔSLR)
QR=TS Opposite sides of a parallelogram are equal
ΔQMRΔSKT Congruent triangles (Angle-Side-Angle)
M1=K1 Congruent triangles have equal angles (ΔQMRΔSKT)
M1=M2 and K1=K2 Vertically opposite angles are equal
JKLM is a parallelogram Opposite angles in a parallelogram are equal

Submit your answer as: andandandandandandand

Angles in a full turn

In the diagram below, line BC is a straight line with angles labelled above and below it.

Answer the following questions about the diagram:

Answer:
  1. x= °
  2. y= °
  3. y is .
numeric
numeric
HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

Angles on a straight line add up to 180°.


STEP: <no title>
[−3 points ⇒ 0 / 3 points left]

Line BC is a straight line, and angles on a straight line up to 180°. This means that all of the angles above BC add up to 180°:

x+56°+89°=180°(s on a str line)x=35°

In the same way, all of the angles below BC add up to 180°:

20°+39°+37°+y=180°(s on a str line)y=84°

Here is the completed diagram, with all of the angle values labelled:

The angles in a full turn (revolution) must add up to 360°. This is a good way to check our answers:

35°+56°+89°+20°+39°+37°+84°=360°

y=84°

y is an acute angle (less than 90°).


Submit your answer as: andand

Angles in a full turn

In the diagram below, line BC is a straight line with angles labelled above and below it.

Answer the following questions about the diagram:

Answer:
  1. x= °
  2. y= °
  3. y is .
numeric
numeric
HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

Angles on a straight line add up to 180°.


STEP: <no title>
[−3 points ⇒ 0 / 3 points left]

Line BC is a straight line, and angles on a straight line up to 180°. This means that all of the angles above BC add up to 180°:

20°+146°+x=180°(s on a str line)x=14°

In the same way, all of the angles below BC add up to 180°:

20°+40°+32°+y=180°(s on a str line)y=88°

Here is the completed diagram, with all of the angle values labelled:

The angles in a full turn (revolution) must add up to 360°. This is a good way to check our answers:

20°+146°+14°+20°+40°+32°+88°=360°

y=88°

y is an acute angle (less than 90°).


Submit your answer as: andand

Angles in a full turn

In the diagram below, line ST is a straight line with angles labelled above and below it.

Answer the following questions about the diagram:

Answer:
  1. x= °
  2. y= °
  3. y is .
numeric
numeric
HINT: <no title>
[−0 points ⇒ 3 / 3 points left]

Angles on a straight line add up to 180°.


STEP: <no title>
[−3 points ⇒ 0 / 3 points left]

Line ST is a straight line, and angles on a straight line up to 180°. This means that all of the angles above ST add up to 180°:

55°+x+13°=180°(s on a str line)x=112°

In the same way, all of the angles below ST add up to 180°:

22°+84°+26°+y=180°(s on a str line)y=48°

Here is the completed diagram, with all of the angle values labelled:

The angles in a full turn (revolution) must add up to 360°. This is a good way to check our answers:

55°+112°+13°+22°+84°+26°+48°=360°

y=48°

y is an acute angle (less than 90°).


Submit your answer as: andand

Relationships between quadrilateral definitions

It is useful to think about quadrilaterals as a connected family. One type of quadrilateral can have all the same properties as another type of quadrilateral, plus some extra properties.

Select from the options to complete the statement about a square.

Answer: A square is a rectangle with:
HINT: <no title>
[−0 points ⇒ 1 / 1 points left]

What is the definition of a rectangle? What additional properties does a square have?


STEP: Compare the definitions for a square and a rectangle
[−1 point ⇒ 0 / 1 points left]
A rectangle is a quadrilateral with two pairs of opposite sides parallel and all corners 90°.
A square is a quadrilateral with two pairs of opposite sides parallel, adjacent sides equal, and all corners 90°.

The following diagram shows a rectangle next to a square, with their defining properties filled in.

We can see that the square has the same properties as the rectangle, plus adjacent sides equal.


Submit your answer as:

Relationships between quadrilateral definitions

It is useful to think about quadrilaterals as a connected family. One type of quadrilateral can have all the same properties as another type of quadrilateral, plus some extra properties.

Select from the options to complete the statement about a rectangle.

Answer: A rectangle is a parallelogram with:
HINT: <no title>
[−0 points ⇒ 1 / 1 points left]

What is the definition of a parallelogram? What additional properties does a rectangle have?


STEP: Compare the definitions for a rectangle and a parallelogram
[−1 point ⇒ 0 / 1 points left]
A parallelogram is a quadrilateral with two pairs of opposite sides parallel.
A rectangle is a quadrilateral with two pairs of opposite sides parallel and all corners 90°.

The following diagram shows a parallelogram next to a rectangle, with their defining properties filled in.

NOTE: It is true that the opposite sides of a rectangle are the same length. But, this is also true for all parallelograms. So, we do not say that a rectangle is a parallelogram with opposite sides equal.

We can see that the rectangle has the same properties as the parallelogram, plus corner angles 90°.


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Relationships between quadrilateral definitions

It is useful to think about quadrilaterals as a connected family. One type of quadrilateral can have all the same properties as another type of quadrilateral, plus some extra properties.

Select from the options to complete the statement about a square.

Answer: A square is a rhombus with:
HINT: <no title>
[−0 points ⇒ 1 / 1 points left]

What is the definition of a rhombus? What additional properties does a square have?


STEP: Compare the definitions for a square and a rhombus
[−1 point ⇒ 0 / 1 points left]
A rhombus is a quadrilateral with two pairs of opposite sides parallel and adjacent sides equal.
A square is a quadrilateral with two pairs of opposite sides parallel, adjacent sides equal, and all corners 90°.

The following diagram shows a rhombus next to a square, with their defining properties filled in.

We can see that the square has the same properties as the rhombus, plus corner angles 90°.


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Using diagonals to prove properties

In the diagram below, JKLM is a quadrilateral with JK=KL and JKN=LKN=29°.

  1. How should we prove that JM^N=LM^N?

    Answer:
    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]

    You must only use the information you have been given. Can you use this information to prove that any triangles are congruent? Then, their matching angles and sides will be equal.


    STEP: Choose which triangles to prove congruent
    [−1 point ⇒ 0 / 1 points left]

    JM^N is a angle in ΔJKM and LM^N is the matching angle in ΔLKM. So, if we prove that ΔJKMΔLKM , then JM^N and LM^N must be equal because congruent triangles are equal in every way.

    We also have to check that we have enough information to prove that ΔJKMΔLKM . See if you can spot two pairs of sides and one pair of angles that are equal between the two triangles!


    Submit your answer as:
  2. The diagram of quadrilateral JKLM is repeated for convenience:

    Now, prove that JM^N=LM^N by selecting the correct options.

    Answer:

    In :

    1. is common.
    2. JKN=LKN (given)

    JM^N=LM^N

    HINT: <no title>
    [−0 points ⇒ 6 / 6 points left]

    Prove congruency by matching up pairs of sides and angles. Use the given information and geometry facts.


    STEP: Complete the proof
    [−6 points ⇒ 0 / 6 points left]
    TIP:
    • Look for any sides or angles that the two triangles have in common.
    • Then, write down any pairs of equal sides or angles that you have been given.
    • Finally, decide what else you need in order to prove that the two triangles are congruent, according to one of the four cases of congruency.

    The correct proof is:

    In ΔJKM and ΔLKM:

    1. KM is common.
    2. JK=KL (given)
    3. JKN=LKN (given)

    ΔJKMΔLKM (SAS)

    JM^N=LM^N (ΔJKMΔLKM)

    NOTE: You can extend this proof to prove that the diagonals of any rhombus bisect all of its vertices.

    Submit your answer as: andandandandand

Using diagonals to prove properties

In the diagram below, PQRS is a quadrilateral with PT=RT,ST=QT, and PRSQ.

  1. How should we prove that PS=PQ?

    Answer:
    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]

    You must only use the information you have been given. Can you use this information to prove that any triangles are congruent? Then, their matching angles and sides will be equal.


    STEP: Choose which triangles to prove congruent
    [−1 point ⇒ 0 / 1 points left]

    PS is a side in ΔPST and PQ is the matching side in ΔPQT. So, if we prove that ΔPSTΔPQT , then PS and PQ must be equal because congruent triangles are equal in every way.

    We also have to check that we have enough information to prove that ΔPSTΔPQT . See if you can spot two pairs of sides and one pair of angles that are equal between the two triangles!


    Submit your answer as:
  2. The diagram of quadrilateral PQRS is repeated for convenience:

    Now, prove that PS=PQ by selecting the correct options.

    Answer:

    In :

    1. is common.
    2. ST=QT
    3. PTS=PTQ= 90° (given)

    PS=PQ

    HINT: <no title>
    [−0 points ⇒ 6 / 6 points left]

    Prove congruency by matching up pairs of sides and angles. Use the given information and geometry facts.


    STEP: Complete the proof
    [−6 points ⇒ 0 / 6 points left]
    TIP:
    • Look for any sides or angles that the two triangles have in common.
    • Then, write down any pairs of equal sides or angles that you have been given.
    • Finally, decide what else you need in order to prove that the two triangles are congruent, according to one of the four cases of congruency.

    The correct proof is:

    In ΔPST and ΔPQT:

    1. PT is common.
    2. ST=QT (given)
    3. PTS=PTQ=90° (given)

    ΔPSTΔPQT (SAS)

    PS=PQ (ΔPSTΔPQT)

    NOTE: You can extend this proof to prove that if the diagonals of any quadrilateral bisect at 90°, then the quadrilateral is a rhombus.

    Submit your answer as: andandandandand

Using diagonals to prove properties

In the diagram below, PQRS is a quadrilateral with PT=RT,ST=QT, and PRSQ.

  1. How should we prove that PS=RQ?

    Answer:
    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]

    You must only use the information you have been given. Can you use this information to prove that any triangles are congruent? Then, their matching angles and sides will be equal.


    STEP: Choose which triangles to prove congruent
    [−1 point ⇒ 0 / 1 points left]

    PS is a side in ΔPST and RQ is the matching side in ΔRQT. So, if we prove that ΔPSTΔRQT , then PS and RQ must be equal because congruent triangles are equal in every way.

    We also have to check that we have enough information to prove that ΔPSTΔRQT . See if you can spot two pairs of sides and one pair of angles that are equal between the two triangles!


    Submit your answer as:
  2. The diagram of quadrilateral PQRS is repeated for convenience:

    Now, prove that PS=RQ by selecting the correct options.

    Answer:

    In :

    1. ST=QT
    2. PTS=RTQ=90° (given)

    PS=RQ

    HINT: <no title>
    [−0 points ⇒ 6 / 6 points left]

    Prove congruency by matching up pairs of sides and angles. Use the given information and geometry facts.


    STEP: Complete the proof
    [−6 points ⇒ 0 / 6 points left]
    TIP:
    • Look for any sides or angles that the two triangles have in common.
    • Then, write down any pairs of equal sides or angles that you have been given.
    • Finally, decide what else you need in order to prove that the two triangles are congruent, according to one of the four cases of congruency.

    The correct proof is:

    In ΔPST and ΔRQT:

    1. PT=RT (given)
    2. ST=QT (given)
    3. PTS=RTQ= 90° (given)

    ΔPSTΔRQT (SAS)

    PS=RQ (ΔPSTΔRQT)

    NOTE: You can extend this proof to prove that if the diagonals of any quadrilateral bisect at 90°, then the quadrilateral is a rhombus.

    Submit your answer as: andandandandand

Proof: is the quadrilateral a parallelogram?

The figure below shows quadrilateral QRST with Q=S=124° and R=T=56°.

The steps and reasons below prove that QRST is a parallelogram. However, there are two steps and two reasons missing. Choose the correct steps and reasons to complete the proof.

Answer:
STEPS REASONS
Q+T=180°
Co-interior angles of parallel lines have a sum of 180°
Given (56°+124°=180°)
RSQT Co-interior angles of parallel lines have a sum of 180°
QRST is a parallelogram
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

You need to start at the top, and go down the proof line by line. Work out the answers as you go down the proof. And stay patient: this question requires a lot of careful thinking!


STEP: Consider the options and choose the correct steps and reasons
[−4 points ⇒ 0 / 4 points left]

The definition of a parallelogram is: "a quadrilateral with both pairs of opposite sides parallel." To prove that the quadrilateral is a parallelogram, we need to show that the opposite sides are parallel to each other. In other words, the proof should follow whatever steps are needed to show that side QR is parallel to side TS and that side RS is parallel to side QT.

This proof uses the angle values to show that adjacent angles must be co-interior angles within parallel lines. The table below explains the various steps of the proof, using colour to tie the explanation to the steps and reasons in the proof.

Explanations

The yellow rows in the table indicate the given information. This is information we can gather directly from the diagram. The size of all the angles are given.

The green rows represent the application of co-interior angles. Co-interior angles inside parallel lines are supplementary (they have a sum of 180°). Here we use this fact in reverse: if the angles are supplementary the lines are parallel.

Once we have shown that the opposite sides are parallel to each other, we can conclude that quadrilateral QRST is a parallelogram. Note that the final step of a proof will always be the fact that you are trying to prove.

Here is the completed proof with the correct steps and reasons.

Steps Reasons
Q+T=180° Given (124°+56°=180°)
QRTS Co-interior angles of parallel lines have a sum of 180°
T+S=180° Given (56°+124°=180°)
RSQT Co-interior angles of parallel lines have a sum of 180°
QRST is a parallelogram Definition of a parallelogram

Submit your answer as: andandand

Proof: is the quadrilateral a parallelogram?

The figure below shows quadrilateral ABCD with A=C=99° and B=D=81°.

The steps and reasons below prove that ABCD is a parallelogram. However, there are two steps and two reasons missing. Choose the correct steps and reasons to complete the proof.

Answer:
STEPS REASONS
A+D=180°
ABDC Co-interior angles of parallel lines have a sum of 180°
D+C=180°
Co-interior angles of parallel lines have a sum of 180°
Definition of a parallelogram
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

You need to start at the top, and go down the proof line by line. Work out the answers as you go down the proof. And stay patient: this question requires a lot of careful thinking!


STEP: Consider the options and choose the correct steps and reasons
[−4 points ⇒ 0 / 4 points left]

The definition of a parallelogram is: "a quadrilateral with both pairs of opposite sides parallel." To prove that the quadrilateral is a parallelogram, we need to show that the opposite sides are parallel to each other. In other words, the proof should follow whatever steps are needed to show that side AB is parallel to side DC and that side BC is parallel to side AD.

This proof uses the angle values to show that adjacent angles must be co-interior angles within parallel lines. The table below explains the various steps of the proof, using colour to tie the explanation to the steps and reasons in the proof.

Explanations

The yellow rows in the table indicate the given information. This is information we can gather directly from the diagram. The size of all the angles are given.

The green rows represent the application of co-interior angles. Co-interior angles inside parallel lines are supplementary (they have a sum of 180°). Here we use this fact in reverse: if the angles are supplementary the lines are parallel.

Once we have shown that the opposite sides are parallel to each other, we can conclude that quadrilateral ABCD is a parallelogram. Note that the final step of a proof will always be the fact that you are trying to prove.

Here is the completed proof with the correct steps and reasons.

Steps Reasons
A+D=180° Given (99°+81°=180°)
ABDC Co-interior angles of parallel lines have a sum of 180°
D+C=180° Given (81°+99°=180°)
BCAD Co-interior angles of parallel lines have a sum of 180°
ABCD is a parallelogram Definition of a parallelogram

Submit your answer as: andandand

Proof: is the quadrilateral a parallelogram?

The figure below shows quadrilateral QRST with Q=S=56° and R=T=124°.

The steps and reasons below prove that QRST is a parallelogram. However, there are two steps and two reasons missing. Choose the correct steps and reasons to complete the proof.

Answer:
STEPS REASONS
Q+T=180° Given (56°+124°=180°)
Co-interior angles of parallel lines have a sum of 180°
T+S=180°
RSQT
Definition of a parallelogram
HINT: <no title>
[−0 points ⇒ 4 / 4 points left]

You need to start at the top, and go down the proof line by line. Work out the answers as you go down the proof. And stay patient: this question requires a lot of careful thinking!


STEP: Consider the options and choose the correct steps and reasons
[−4 points ⇒ 0 / 4 points left]

The definition of a parallelogram is: "a quadrilateral with both pairs of opposite sides parallel." To prove that the quadrilateral is a parallelogram, we need to show that the opposite sides are parallel to each other. In other words, the proof should follow whatever steps are needed to show that side QR is parallel to side TS and that side RS is parallel to side QT.

This proof uses the angle values to show that adjacent angles must be co-interior angles within parallel lines. The table below explains the various steps of the proof, using colour to tie the explanation to the steps and reasons in the proof.

Explanations

The yellow rows in the table indicate the given information. This is information we can gather directly from the diagram. The size of all the angles are given.

The green rows represent the application of co-interior angles. Co-interior angles inside parallel lines are supplementary (they have a sum of 180°). Here we use this fact in reverse: if the angles are supplementary the lines are parallel.

Once we have shown that the opposite sides are parallel to each other, we can conclude that quadrilateral QRST is a parallelogram. Note that the final step of a proof will always be the fact that you are trying to prove.

Here is the completed proof with the correct steps and reasons.

Steps Reasons
Q+T=180° Given (56°+124°=180°)
QRTS Co-interior angles of parallel lines have a sum of 180°
T+S=180° Given (124°+56°=180°)
RSQT Co-interior angles of parallel lines have a sum of 180°
QRST is a parallelogram Definition of a parallelogram

Submit your answer as: andandand

Relationships between quadrilaterals: true or false

Consider the following statement and decide whether it is true or false.

All squares are parallelograms.
Answer: The statement is .
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Think about the definitions of the quadrilaterals and how they relate to each other. Does a square always have at least the same properties as a parallelogram? Or, can you think of any squares that are not types of parallelograms?


STEP: Think about the definitions of the quadrilaterals
[−2 points ⇒ 0 / 2 points left]
A square is a quadrilateral with two pairs of opposite sides that are parallel, equal adjacent sides, and corners that are 90°.
A parallelogram is a quadrilateral with two pairs of opposite sides that are parallel.

We can see that a square has all of the same properties as a parallelogram (and some more). So, squares are special types of parallelograms. Therefore, the statement is true.

NOTE: It does not matter that a square has extra properties that a parallelogram does not have. What matters is that it has at least the properties of a parallelogram.

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Relationships between quadrilaterals: true or false

Consider the following statement and decide whether it is true or false.

Some rhombuses are squares.
Answer: The statement is .
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Think about the definitions of the quadrilaterals and how they relate to each other. Does a rhombus always have at least the same properties as a square? Or, can you think of any rhombuses that are not types of squares?


STEP: Think about the definitions of the quadrilaterals
[−2 points ⇒ 0 / 2 points left]

The family of quadrilaterals called rhombuses all have two pairs of opposite sides that are parallel and equal adjacent sides.

Within this family there is a group that has these properties plus corner angles that are equal to 90°. These special rhombuses are called squares.

Therefore, some rhombuses are squares, and the statement is true.

NOTE: If we had claimed that all rhombuses are squares, this would have been false. But for this statement to be true we only need some rhombuses to be squares.

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Relationships between quadrilaterals: true or false

Consider the following statement and decide whether it is true or false.

All kites are rhombuses.
Answer: The statement is .
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Think about the definitions of the quadrilaterals and how they relate to each other. Does a kite always have at least the same properties as a rhombus? Or, can you think of any kites that are not types of rhombuses?


STEP: Think about the definitions of the quadrilaterals
[−2 points ⇒ 0 / 2 points left]
A kite is a quadrilateral with two pairs of equal adjacent sides.
A rhombus is a quadrilateral with two pairs of opposite sides that are parallel and equal adjacent sides.

Consider the following kite:

This shape meets the definition of a kite. But, it does not meet the definition of a rhombus because the opposite sides are not parallel. So, we have come up with an example that disagrees with the statement. Therefore, the statement is false.

NOTE: An example that disagrees with the statement is called a counter-example. We only need one counter-example to prove that a statement is false. This is because if it is false for one example, then we cannot say that it is true for all - in other words, it is false.

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Proving diagonals using congruency

In the diagram below, ABCD is a quadrilateral with AD=BC and AB^C=BA^D= 90°.

Prove that AC=BD.

  1. How should we prove that AC=BD?

    Answer:
    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]

    You must only use the information you have been given. Can you use this information to prove that any triangles are congruent? Then, their matching angles and sides will be equal.


    STEP: Choose which triangles to prove congruent
    [−1 point ⇒ 0 / 1 points left]

    The correct strategy is: Prove that ΔABDΔBAC.

    AC is a side in ΔBAC and BD is the matching side in ΔABD. So, if we prove that ΔABDΔBAC, then AC and BD must be equal because congruent triangles are equal in every way.

    We also have to check that we have enough information to prove that ΔABDΔBAC. See if you can spot two pairs of sides and one pair of angles that are equal between the two triangles!


    Submit your answer as:
  2. The diagram of quadrilateral ABCD is repeated for convenience:

    Now, prove that AC=BD by selecting the correct options.

    Answer:

    In :

    1. is common.
    2. AD=BC
    3. ABC=DAB=90° (given)


    AC=BD

    HINT: <no title>
    [−0 points ⇒ 6 / 6 points left]

    Prove congruency by matching up pairs of sides and angles. Use the given information and geometry facts.


    STEP: Complete the proof
    [−6 points ⇒ 0 / 6 points left]
    TIP:
    • Look for any sides or angles that the two triangles have in common.
    • Then, write down any pairs of equal sides or angles that you have been given.
    • Finally, decide what else you need in order to prove that the two triangles are congruent, according to one of the four cases of congruency.

    The correct proof is:

    In ΔABD and ΔBAC :

    1. AB is common.
    2. AD=BC (given)
    3. ABC=DAB=90° (given)

    ΔABDΔBAC (SAS)

    AC=BD (ΔABDΔBAC)

    NOTE: You can extend this proof to prove that diagonals are equal in all rectangles. This is one of the properties that you should know already. Now you know why it is true!

    Submit your answer as: andandandandand

Proving diagonals using congruency

In the diagram below, WVXY is a quadrilateral with WV=XV and WZ=XZ .

Prove that WZ^V=90°.

  1. How should we prove that WZ^V=90°?

    Answer:
    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]

    You must only use the information you have been given. Can you use this information to prove that any triangles are congruent? Then, their matching angles and sides will be equal.


    STEP: Choose which triangles to prove congruent
    [−1 point ⇒ 0 / 1 points left]

    The correct strategy is: Prove that ΔWZVΔXZV.

    If we can prove that ΔWZVΔXZV, then we can prove that WZ^V=XZ^V. This is because they are matching angles in the two triangles, and congruent triangles are equal in every way.

    And, WZ^V+XZ^V=180° (s on a str line).

    But since the two angles are equal, this means both angles must be 90°.

    We also have to check that we have enough information to prove that ΔWZVΔXZV. See if you can spot three pairs of sides that are equal between the two triangles!


    Submit your answer as:
  2. The diagram of quadrilateral WVXY is repeated for convenience:

    Now, prove that WZ^V=90° by selecting the correct options.

    Answer:

    In :

    1. is common.
    2. WV=XV (given)
    3. WZ=XZ (given)

    WZV=XZV
    WZV=XZV=90°

    HINT: <no title>
    [−0 points ⇒ 6 / 6 points left]

    Prove congruency by matching up pairs of sides and angles. Use the given information and geometry facts.


    STEP: Complete the proof
    [−6 points ⇒ 0 / 6 points left]
    TIP:
    • Look for any sides or angles that the two triangles have in common.
    • Then, write down any pairs of equal sides or angles that you have been given.
    • Finally, decide what else you need in order to prove that the two triangles are congruent, according to one of the four cases of congruency.

    The correct proof is:

    In ΔWZV and ΔXZV :

    1. ZV is common.
    2. WV=XV (given)
    3. WZ=XZ (given)

    ΔWZVΔXZV (SSS)

    AEB=CEB (ΔAEBΔCEB)
    WZV=XZV=90° (s on a str line)

    NOTE: You can extend this proof to prove that diagonals are perpendicular in all rhombuses. This is one of the properties that you should know already. Now you know why it is true!

    Submit your answer as: andandandandand

Proving diagonals using congruency

In the diagram below, WVXY is a quadrilateral with WY=VX and WV^X=VW^Y= 90°.

Prove that WX=VY.

  1. How should we prove that WX=VY?

    Answer:
    HINT: <no title>
    [−0 points ⇒ 1 / 1 points left]

    You must only use the information you have been given. Can you use this information to prove that any triangles are congruent? Then, their matching angles and sides will be equal.


    STEP: Choose which triangles to prove congruent
    [−1 point ⇒ 0 / 1 points left]

    The correct strategy is: Prove that ΔWVYΔVWX.

    WX is a side in ΔVWX and VY is the matching side in ΔWVY. So, if we prove that ΔWVYΔVWX, then WX and VY must be equal because congruent triangles are equal in every way.

    We also have to check that we have enough information to prove that ΔWVYΔVWX. See if you can spot two pairs of sides and one pair of angles that are equal between the two triangles!


    Submit your answer as:
  2. The diagram of quadrilateral WVXY is repeated for convenience:

    Now, prove that WX=VY by selecting the correct options.

    Answer:

    In :

    1. is common.
    2. WY=VX
    3. WVX=YWV=90° (given)


    WX=VY

    HINT: <no title>
    [−0 points ⇒ 6 / 6 points left]

    Prove congruency by matching up pairs of sides and angles. Use the given information and geometry facts.


    STEP: Complete the proof
    [−6 points ⇒ 0 / 6 points left]
    TIP:
    • Look for any sides or angles that the two triangles have in common.
    • Then, write down any pairs of equal sides or angles that you have been given.
    • Finally, decide what else you need in order to prove that the two triangles are congruent, according to one of the four cases of congruency.

    The correct proof is:

    In ΔWVY and ΔVWX :

    1. WV is common.
    2. WY=VX (given)
    3. WVX=YWV=90° (given)

    ΔWVYΔVWX (SAS)

    WX=VY (ΔWVYΔVWX)

    NOTE: You can extend this proof to prove that diagonals are equal in all rectangles. This is one of the properties that you should know already. Now you know why it is true!

    Submit your answer as: andandandandand

Calculate angles between parallel lines

In the diagram below, FG HJ. KL is a transversal.

NOTE: A transversal is a line that cuts across two other lines.

What is the value of x?

Answer: x= °
numeric
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

STEP: <no title>
[−2 points ⇒ 0 / 2 points left]

Alternate angles are both inside the parallel lines, but on opposite sides of the transversal. If we highlight the angles we will see a "Z" shape. This helps us to identify alternate angles.

Since we know that alternate angles are equal, we can write:

x=74°(alt s,FGHJ)

Therefore x=74° (alt s, FGHJ).

NOTE: Although you didn't need to give a reason here, reasons are important in geometry. You should learn these reasons and always write them in your work.

Submit your answer as:

Calculate angles between parallel lines

In the diagram below, EF GH. JK is a transversal.

NOTE: A transversal is a line that cuts across two other lines.

What is the value of x?

Answer: x= °
numeric
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

STEP: <no title>
[−2 points ⇒ 0 / 2 points left]

Co-interior angles are both inside the parallel lines, on the same side of the transversal. If we highlight the angles we will see a "U" shape. This helps us to identify co-interior angles.

Since we know that co-interior angles add up to 180°, we can write:

x+100°=180°(co-int s,EFGH)x=180°100°x=80°

Therefore x=80° (co-int s, EFGH).

NOTE: Although you didn't need to give a reason here, reasons are important in geometry. You should learn these reasons and always write them in your work.

Submit your answer as:

Calculate angles between parallel lines

In the diagram below, CD EF. GH is a transversal.

NOTE: A transversal is a line that cuts across two other lines.

What is the value of x?

Answer: x= °
numeric
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

STEP: <no title>
[−2 points ⇒ 0 / 2 points left]

Co-interior angles are both inside the parallel lines, on the same side of the transversal. If we highlight the angles we will see a "U" shape. This helps us to identify co-interior angles.

Since we know that co-interior angles add up to 180°, we can write:

x+117°=180°(co-int s,CDEF)x=180°117°x=63°

Therefore x=63° (co-int s, CDEF).

NOTE: Although you didn't need to give a reason here, reasons are important in geometry. You should learn these reasons and always write them in your work.

Submit your answer as:

Given diagonal properties, identify the quadrilateral

Consider the following information about the diagonals of a quadrilateral PQRS :

The diagonals of PQRS are equal in length and bisect the corner angles.

Select the type of quadrilateral for which this statement is definitely true.

Answer: PQRS is a:
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Think about the diagonal properties of the different quadrilaterals. Which one has the exact properties described in the question?


STEP: Recall diagonal properties
[−2 points ⇒ 0 / 2 points left]

The diagonals of all rectangles are equal in length.

The diagonals of all rhombuses bisect the corner angles.

NOTE: Bisect means to cut into two equal parts.

So PQRS must be a type of rectangle, and also a type of rhombus. So, PQRS must be a square.


Submit your answer as:

Given diagonal properties, identify the quadrilateral

Consider the following information about the diagonals of a quadrilateral PQRS :

The diagonals of PQRS are perpendicular to each other.

Select the type of quadrilateral for which this statement is definitely true.

Answer: PQRS is a:
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Think about the diagonal properties of the different quadrilaterals. Which one has the exact properties described in the question?


STEP: Recall diagonal properties
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Perpendicular means that the diagonals meet at 90°. This is true for all kites.

NOTE: This is true for all kites, so it must also be true for rhombuses and squares (which are types of kites).

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Given diagonal properties, identify the quadrilateral

Consider the following information about the diagonals of a quadrilateral PQRS :

The diagonals of PQRS bisect the corner angles.

Select the type of quadrilateral for which this statement is definitely true.

Answer: PQRS is a:
HINT: <no title>
[−0 points ⇒ 2 / 2 points left]

Think about the diagonal properties of the different quadrilaterals. Which one has the exact properties described in the question?


STEP: Recall diagonal properties
[−2 points ⇒ 0 / 2 points left]

The diagonals of all rhombuses bisect the corner angles.

NOTE: Bisect means to cut into two equal parts.

NOTE: This is true for all rhombuses, so it must also be true for squares (which are types of rhombuses).

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Recognise pairs of angles between parallel lines

In the diagram below, XYTU.  VW is a transversal.

NOTE: A transversal is a line that cuts across two other lines.

What type of angle pair is shown by the coloured angles?

Answer: The angles are .
HINT: <no title>
[−0 points ⇒ 1 / 1 points left]

Can you identify the relationship between the coloured angles?


STEP: Identify the type of angle pair
[−1 point ⇒ 0 / 1 points left]

This intersection creates co-interior angles.

Co-interior angles are both inside the parallel lines, on the same side of the transversal. If we highlight the angles we will see a "U" shape. This helps us identify co-interior angles.


Submit your answer as:

Recognise pairs of angles between parallel lines

In the diagram below, RSTO.  MN is a transversal.

NOTE: A transversal is a line that cuts across two other lines.

What type of angle pair is shown by the coloured angles?

Answer: The angles are .
HINT: <no title>
[−0 points ⇒ 1 / 1 points left]

Can you identify the relationship between the coloured angles?


STEP: Identify the type of angle pair
[−1 point ⇒ 0 / 1 points left]

This intersection creates corresponding angles.

Corresponding angles are in the same position compared to the parallel lines and the transversal. If we highlight the angles we will see an "F" shape. This helps us to identify corresponding angles.


Submit your answer as:

Recognise pairs of angles between parallel lines

In the diagram below, ABCD.  EF is a transversal.

NOTE: A transversal is a line that cuts across two other lines.

What type of angle pair is shown by the coloured angles?

Answer: The angles are .
HINT: <no title>
[−0 points ⇒ 1 / 1 points left]

Can you identify the relationship between the coloured angles?


STEP: Identify the type of angle pair
[−1 point ⇒ 0 / 1 points left]

This intersection creates co-interior angles.

Co-interior angles are both inside the parallel lines, on the same side of the transversal. If we highlight the angles we will see a "U" shape. This helps us identify co-interior angles.


Submit your answer as:

Congruency in kites

  1. Consider the following diagram:

    Prove that ΔLKJΔLMJ by completing the proof below.

    Answer:

    In ΔLKJ and ΔLMJ:

    1. LK=
    2. =MJ
    3. is common

    ΔLKJΔLMJ

    HINT: <no title>
    [−0 points ⇒ 6 / 6 points left]

    Use the properties of a kite to work out which sides are equal to each other.


    STEP: Use the properties of a kite to match up equal sides
    [−6 points ⇒ 0 / 6 points left]

    First, we focus on the two triangles that we need to prove congruent, and ignore the rest of the diagram.

    A kite is a quadrilateral that has two pairs of adjacent sides equal. Using this, and the information we were given in the diagram, we will highlight the sides which we know must be equal in our two triangles.

    In ΔLKJ and ΔLMJ:

    1. LK=LM (adjacent sides of kite)
    2. KJ=MJ (adjacent sides of kite)
    3. LJ is common

    We have proved that three pairs of sides are equal, so we can use the congruency case SSS.

    ΔLKJΔLMJ (SSS)


    Submit your answer as: andandandandand
  2. Consider the following diagram:

    You have just proved that ΔLKJΔLMJ. Hence, determine the size of KJ^L.

    Answer: KJ^L= °
    numeric
    HINT: <no title>
    [−0 points ⇒ 2 / 2 points left]

    Once you have proved congruency, you know that matching angles will be equal. Which angle matches KJ^L?


    STEP: Identify matching angles to determine size
    [−2 points ⇒ 0 / 2 points left]

    We have proved that ΔLKJΔLMJ.

    Therefore, the matching angles in these two triangles must be equal.

    In particular, KJ^L=LJ^M.

    They are only equal because the triangles are congruent, so we use the congruency statement as our reason.

    KJ^L= 25° (ΔLKJΔLMJ)


    Submit your answer as: and

Congruency in kites

  1. Consider the following diagram:

    Prove that ΔJKLΔJML by completing the proof below.

    Answer:

    In ΔJKL and ΔJML:

    1. JK=
    2. =ML
    3. is common

    ΔJKLΔJML

    HINT: <no title>
    [−0 points ⇒ 6 / 6 points left]

    Use the properties of a kite to work out which sides are equal to each other.


    STEP: Use the properties of a kite to match up equal sides
    [−6 points ⇒ 0 / 6 points left]

    First, we focus on the two triangles that we need to prove congruent, and ignore the rest of the diagram.

    A kite is a quadrilateral that has two pairs of adjacent sides equal. Using this, and the information we were given in the diagram, we will highlight the sides which we know must be equal in our two triangles.

    In ΔJKL and ΔJML:

    1. JK=JM (adjacent sides of kite)
    2. KL=ML (adjacent sides of kite)
    3. JL is common

    We have proved that three pairs of sides are equal, so we can use the congruency case SSS.

    ΔJKLΔJML (SSS)


    Submit your answer as: andandandandand
  2. Consider the following diagram:

    You have just proved that ΔJKLΔJML. Hence, determine the size of MJ^L.

    Answer: MJ^L= °
    numeric
    HINT: <no title>
    [−0 points ⇒ 2 / 2 points left]

    Once you have proved congruency, you know that matching angles will be equal. Which angle matches MJ^L?


    STEP: Identify matching angles to determine size
    [−2 points ⇒ 0 / 2 points left]

    We have proved that ΔJKLΔJML.

    Therefore, the matching angles in these two triangles must be equal.

    In particular, MJ^L=KJ^L.

    They are only equal because the triangles are congruent, so we use the congruency statement as our reason.

    MJ^L= 51° (ΔJKLΔJML)


    Submit your answer as: and

Congruency in kites

  1. Consider the following diagram:

    Prove that ΔVWZΔVYZ by completing the proof below.

    Answer:

    In ΔVWZ and ΔVYZ:

    1. VW=
    2. =YZ
    3. is common

    ΔVWZΔVYZ

    HINT: <no title>
    [−0 points ⇒ 6 / 6 points left]

    Use the properties of a kite to work out which sides are equal to each other.


    STEP: Use the properties of a kite to match up equal sides
    [−6 points ⇒ 0 / 6 points left]

    First, we focus on the two triangles that we need to prove congruent, and ignore the rest of the diagram.

    A kite is a quadrilateral that has two pairs of adjacent sides equal. Using this, and the information we were given in the diagram, we will highlight the sides which we know must be equal in our two triangles.

    In ΔVWZ and ΔVYZ:

    1. VW=VY (adjacent sides of kite)
    2. WZ=YZ (given)
    3. VZ is common

    We have proved that three pairs of sides are equal, so we can use the congruency case SSS.

    ΔVWZΔVYZ (SSS)


    Submit your answer as: andandandandand
  2. Consider the following diagram:

    You have just proved that ΔVWZΔVYZ. Hence, determine the size of WV^Z.

    Answer: WV^Z= °
    numeric
    HINT: <no title>
    [−0 points ⇒ 2 / 2 points left]

    Once you have proved congruency, you know that matching angles will be equal. Which angle matches WV^Z?


    STEP: Identify matching angles to determine size
    [−2 points ⇒ 0 / 2 points left]

    We have proved that ΔVWZΔVYZ.

    Therefore, the matching angles in these two triangles must be equal.

    In particular, WV^Z=YV^Z.

    They are only equal because the triangles are congruent, so we use the congruency statement as our reason.

    WV^Z= 47° (ΔVWZΔVYZ)


    Submit your answer as: and

4. Solving riders

Algebraic triangles

Determine the size of z, giving reasons for each of your statements.

INSTRUCTION: There is sometimes more than one way of solving a geometry problem. In this question, you must follow the structure given to you below. You should start by giving the answer for C^ in terms of z.
Answer: C^=
z= °
expression
numeric
HINT: <no title>
[−0 points ⇒ 5 / 5 points left]

What do you know about triangles that have two equal sides? Once you've found C^ in terms of z, write an equation for this triangle.


STEP: Find a missing angle in terms of z
[−2 points ⇒ 3 / 5 points left]

The triangle has two equal sides, so we know that the angles opposite those sides will be equal.

C^=3z(s opp equal sides)

STEP: Write an equation for the triangle
[−2 points ⇒ 1 / 5 points left]

We know all the angles of the triangle in terms of z, so we can write an equation using the sum of angles inside the triangle.

3z+3z+5z40°=180°(sum of s in Δ)

STEP: Solve the equation to determine the value of z
[−1 point ⇒ 0 / 5 points left]
11z40°=180°11z40°+40°=180°+40°11z=220°11z11=220°11z=20°

Submit your answer as: andandand

Exercises

Algebraic triangles

Determine the size of z, giving reasons for each of your statements.

INSTRUCTION: There is sometimes more than one way of solving a geometry problem. In this question, you must follow the structure given to you below. You should start by giving the answer for Q^ in terms of z.
Answer: Q^=
z= °
expression
numeric
HINT: <no title>
[−0 points ⇒ 5 / 5 points left]

What do you know about triangles that have two equal sides? Once you've found Q^ in terms of z, write an equation for this triangle.


STEP: Find a missing angle in terms of z
[−2 points ⇒ 3 / 5 points left]

The triangle has two equal sides, so we know that the angles opposite those sides will be equal.

Q^=3z(s opp equal sides)

STEP: Write an equation for the triangle
[−2 points ⇒ 1 / 5 points left]

We know the two opposite interior angles in terms of z, so we can write an equation using the exterior angle of this triangle.

3z+3z=4z+38°(ext  of Δ)

STEP: Solve the equation to determine the value of z
[−1 point ⇒ 0 / 5 points left]
6z=4z+38°6z4z=4z+38°4z2z=38°2z2=38°2z=19°

Submit your answer as: andandand

Algebraic triangles

Determine the size of x, giving reasons for each of your statements.

INSTRUCTION: There is sometimes more than one way of solving a geometry problem. In this question, you must follow the structure given to you below. You should start by giving the answer for R^ in terms of x.
Answer: R^=
x= °
expression
numeric
HINT: <no title>
[−0 points ⇒ 5 / 5 points left]

What do you know about triangles that have two equal sides? Once you've found R^ in terms of x, write an equation for this triangle.


STEP: Find a missing angle in terms of x
[−2 points ⇒ 3 / 5 points left]

The triangle has two equal sides, so we know that the angles opposite those sides will be equal.

R^=4x(s opp equal sides)

STEP: Write an equation for the triangle
[−2 points ⇒ 1 / 5 points left]

We know all the angles of the triangle in terms of x, so we can write an equation using the sum of angles inside the triangle.

4x+4x+4x+12°=180°(sum of s in Δ)

STEP: Solve the equation to determine the value of x
[−1 point ⇒ 0 / 5 points left]
12x+12°=180°12x+12°12°=180°12°12x=168°12x12=168°12x=14°

Submit your answer as: andandand

Algebraic triangles

Determine the size of y, giving reasons for each of your statements.

INSTRUCTION: There is sometimes more than one way of solving a geometry problem. In this question, you must follow the structure given to you below. You should start by giving the answer for Q^ in terms of y.
Answer: Q^=
y= °
expression
numeric
HINT: <no title>
[−0 points ⇒ 5 / 5 points left]

What do you know about triangles that have two equal sides? Once you've found Q^ in terms of y, write an equation for this triangle.


STEP: Find a missing angle in terms of y
[−2 points ⇒ 3 / 5 points left]

The triangle has two equal sides, so we know that the angles opposite those sides will be equal.

Q^=2y(s opp equal sides)

STEP: Write an equation for the triangle
[−2 points ⇒ 1 / 5 points left]

We know the two opposite interior angles in terms of y, so we can write an equation using the exterior angle of this triangle.

2y+2y=3y+26°(ext  of Δ)

STEP: Solve the equation to determine the value of y
[−1 point ⇒ 0 / 5 points left]
4y=3y+26°4y3y=3y+26°3yy=26°

Submit your answer as: andandand